Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools

Calculus Formulas

Limits
Derivatives of Trigonometric Functions
Integrals
Fundamental Theorem of Calculus
Limit Laws
Derivatives of Inverse Trigonometric Functions
Integrals of Trigonometric Functions
Techniques of Integration
Special Limits
Derivatives of Exponential and Logarithmic Functions
Basic Integration Rules
Derivatives
Basic Differentiation Rules
Integrals of Exponential and Logarithmic Functions

Limits






Definition Of A Limit



Formula:

lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
This means that as x gets super close to 'a', the function f(x) gets super close to 'L'. Think of it like f(x) is chasing L as x approaches a.
x is getting closer to the value 'a'
The function we're interested in
The value that f(x) is approaching
If f(x)=2xf(x) = 2x, then lim⁡x→3f(x)=6\lim_{x \to 3} f(x) = 6 because as x gets close to 3, f(x) gets close to 6.
Understanding how functions behave near a specific point, even if they're not defined exactly at that point.





Limit Of A Constant Function



Formula:

lim⁡x→ac=c\lim_{x \to a} c = c
No matter where x goes, a constant function stays the same. So the limit is just that constant.
A constant value
If c=4c = 4, then lim⁡x→104=4\lim_{x \to 10} 4 = 4
Simplifying limits involving constants.





Limit Of Identity Function



Formula:

lim⁡x→ax=a\lim_{x \to a} x = a
As x approaches a, well, x just becomes a. There's nothing tricky here.
The variable itself
The value x is approaching
lim⁡x→5x=5\lim_{x \to 5} x = 5
Simplifying basic limits.

Limit Laws






Limit Laws - Sum Rule



Formula:

lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
You can split the limit of a sum into the sum of the limits. It's like handling each function separately and then adding the results.
Functions we're adding together
Limit of f(x) as x approaches a
Limit of g(x) as x approaches a
If lim⁡x→2f(x)=5\lim_{x \to 2} f(x) = 5 and lim⁡x→2g(x)=3\lim_{x \to 2} g(x) = 3, then lim⁡x→2[f(x)+g(x)]=5+3=8\lim_{x \to 2} [f(x) + g(x)] = 5 + 3 = 8
Breaking down complex limits into simpler parts to make calculations easier.





Limit Laws - Difference Rule



Formula:

lim⁡x→a[f(x)−g(x)]=lim⁡x→af(x)−lim⁡x→ag(x)\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)
Just like addition, you can split the limit over subtraction. Handle each part separately, then subtract the results.
Functions we're subtracting
Limit of f(x) as x approaches a
Limit of g(x) as x approaches a
If lim⁡x→4f(x)=10\lim_{x \to 4} f(x) = 10 and lim⁡x→4g(x)=6\lim_{x \to 4} g(x) = 6, then lim⁡x→4[f(x)−g(x)]=10−6=4\lim_{x \to 4} [f(x) - g(x)] = 10 - 6 = 4
Simplifying limits involving subtraction.





Limit Laws - Product Rule



Formula:

lim⁡x→a[f(x)⋅g(x)]=(lim⁡x→af(x))⋅(lim⁡x→ag(x))\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)
The limit of a product is the product of the limits. Calculate each limit separately, then multiply them.
Functions we're multiplying
Limit of f(x)
Limit of g(x)
If lim⁡x→1f(x)=2\lim_{x \to 1} f(x) = 2 and lim⁡x→1g(x)=5\lim_{x \to 1} g(x) = 5, then lim⁡x→1[f(x)⋅g(x)]=2⋅5=10\lim_{x \to 1} [f(x) \cdot g(x)] = 2 \cdot 5 = 10
Breaking down complex multiplication within limits.





Limit Laws - Quotient Rule



Formula:

lim⁡x→a(f(x)g(x))=lim⁡x→af(x)lim⁡x→ag(x)\lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, provided lim⁡x→ag(x)≠0\lim_{x \to a} g(x) \ne 0
The limit of a fraction is the fraction of the limits, as long as you're not dividing by zero.
Numerator function
Denominator function
Denominator's limit isn't zero
If lim⁡x→2f(x)=8\lim_{x \to 2} f(x) = 8 and lim⁡x→2g(x)=4\lim_{x \to 2} g(x) = 4, then lim⁡x→2(f(x)g(x))=84=2\lim_{x \to 2} \left( \frac{f(x)}{g(x)} \right) = \frac{8}{4} = 2
Simplifying limits involving division.





Limit Laws - Constant Multiple Rule



Formula:

lim⁡x→a[c⋅f(x)]=c⋅lim⁡x→af(x)\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)
If you're multiplying a function by a constant, you can pull the constant out of the limit.
A constant number
Function being multiplied
If c=3c = 3 and lim⁡x→1f(x)=7\lim_{x \to 1} f(x) = 7, then lim⁡x→1[3⋅f(x)]=3⋅7=21\lim_{x \to 1} [3 \cdot f(x)] = 3 \cdot 7 = 21
Simplifying limits when constants are involved.

Special Limits






Special Limit Of Sine Over X



Formula:

lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1
As x gets tiny (close to zero), sin(x) behaves almost exactly like x. So their ratio approaches 1.
x is approaching zero
Sine of x
For x = 0.01 radians, sin⁡0.010.01≈0.99998\frac{\sin 0.01}{0.01} \approx 0.99998
Evaluating limits in trigonometric functions, especially in derivatives.





Special Exponential Limit



Formula:

lim⁡x→0ex−1x=1\lim_{x \to 0} \frac{e^x - 1}{x} = 1
When x is very small, e^x is approximately 1 + x. So the numerator is about x, and the ratio approaches 1.
Exponential function with base e
x is approaching zero
For x = 0.001, e0.001−10.001≈1\frac{e^{0.001} - 1}{0.001} \approx 1
Finding derivatives of exponential functions.





Limit Of (1 + 1/n)^n



Formula:

lim⁡n→∞(1+1n)n=e\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
As n gets huge, this expression gets closer and closer to e (about 2.71828). It's like the magic formula that defines e.
n is growing without bound
The natural exponential base
For n = 1,000, (1+11000)1000≈2.7169\left(1 + \frac{1}{1000}\right)^{1000} \approx 2.7169
Understanding continuous growth, compound interest.

Derivatives






Definition Of The Derivative



Formula:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
This formula gives us the exact slope of the function f(x) at any point x. Think of h as a tiny step towards x; as h approaches zero, we get the instantaneous rate of change.
Derivative of f at x (slope at point x)
h is getting infinitesimally small
Change in function's value over interval h
For f(x)=x2f(x) = x^2, the derivative is f′(x)=lim⁡h→0(x+h)2−x2h=2xf'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} = 2x
Finding instantaneous rates of change, slopes of tangent lines.

Basic Differentiation Rules






Power Rule



Formula:

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}
When differentiating x raised to a power, you bring down the exponent in front and subtract one from the exponent. It's a quick way to find slopes of polynomial functions.
Variable x raised to exponent n
Exponent (any real number)
Derivative of x^n
For f(x)=x3f(x) = x^3, f′(x)=3x2f'(x) = 3x^{2}
Differentiating polynomials, calculating velocities from position functions.





Constant Rule



Formula:

ddx[c]=0\frac{d}{dx}[c] = 0
The derivative of any constant is zero because constants don't change, so their rate of change is zero.
A constant number
The derivative of any constant
If f(x)=5f(x) = 5, then f′(x)=0f'(x) = 0
Simplifying derivatives of functions that include constant terms.





Constant Multiple Rule



Formula:

ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)
If you're multiplying a function by a constant, you can pull the constant out and multiply it by the derivative of the function.
A constant multiplier
Function being multiplied
Derivative after applying the rule
If f(x)=3x2f(x) = 3x^2, then f′(x)=3⋅2x=6xf'(x) = 3 \cdot 2x = 6x
Simplifying derivatives when constants are involved.





Sum And Difference Rules



Formula:

ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)
The derivative of a sum or difference is just the sum or difference of the derivatives. Differentiate each function separately, then add or subtract them.
Functions being added or subtracted
Derivatives of the functions
If f(x)=x2+3xf(x) = x^2 + 3x, then f′(x)=2x+3f'(x) = 2x + 3
Simplifying derivatives of combined functions.





Product Rule



Formula:

ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)
When differentiating a product of two functions, you take the derivative of the first times the second, plus the first times the derivative of the second.
Functions being multiplied
Derivatives of the functions
If f(x)=x⋅sin⁡xf(x) = x \cdot \sin x, then f′(x)=1⋅sin⁡x+x⋅cos⁡xf'(x) = 1 \cdot \sin x + x \cdot \cos x
Differentiating products of functions, like in physics for work calculations.





Quotient Rule



Formula:

ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
To differentiate a quotient, use this formula: bottom times derivative of the top minus top times derivative of the bottom, all over the bottom squared.
Numerator function
Denominator function
Derivatives of the functions
Square of the denominator function
If f(x)=x2x+1f(x) = \frac{x^2}{x+1}, then f′(x)=(2x)(x+1)−x2(1)(x+1)2f'(x) = \frac{(2x)(x+1) - x^2(1)}{(x+1)^2}
Differentiating ratios of functions, important in rate problems.





Chain Rule



Formula:

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
When differentiating a composite function, you take the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Think of it as peeling back layers.
Composite function (function within a function)
Derivative of the outer function evaluated at inner function
Derivative of the inner function
If f(x)=(3x+2)5f(x) = (3x + 2)^5, then f′(x)=5(3x+2)4⋅3f'(x) = 5(3x + 2)^{4} \cdot 3
Differentiating complex functions like nested functions or functions raised to a power.

Derivatives Of Trigonometric Functions






Derivative Of Sine Function



Formula:

ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x
The rate at which sin x changes with respect to x is cos x. It shows how the slope of the sine curve at any point x is given by the cosine of that x.
Sine function
Cosine function
If f(x)=sin⁡xf(x) = \sin x, then f′(x)=cos⁡xf'(x) = \cos x
Analyzing oscillations, waves, and circular motion.





Derivative Of Cosine Function



Formula:

ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x
The derivative of cos x is negative sin x. This tells us the slope of the cosine curve at any point x is the negative sine of x.
Cosine function
Negative sine function
If f(x)=cos⁡xf(x) = \cos x, then f′(x)=−sin⁡xf'(x) = -\sin x
Studying harmonic motion, electrical circuits.





Derivative Of Tangent Function



Formula:

ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x
The derivative of tan x is secant squared x. This shows how rapidly the tangent function changes at any point x.
Tangent function
Secant squared function
If f(x)=tan⁡xf(x) = \tan x, then f′(x)=sec⁡2xf'(x) = \sec^2 x
Calculating slopes of tangent lines, optics.

Derivatives Of Exponential And Logarithmic Functions






Derivative Of Exponential Function



Formula:

ddx[ex]=ex\frac{d}{dx}[e^{x}] = e^{x}
The derivative of e to the x is just e to the x. The exponential function grows at a rate proportional to its value.
Exponential function with base e
If f(x)=exf(x) = e^{x}, then f′(x)=exf'(x) = e^{x}
Modeling continuous growth or decay, compound interest.





Derivative Of A^x



Formula:

ddx[ax]=axln⁡a\frac{d}{dx}[a^{x}] = a^{x} \ln a
When differentiating an exponential function with base a, multiply the original function by the natural log of the base.
Exponential function with base a
Natural logarithm of the base a
If f(x)=2xf(x) = 2^{x}, then f′(x)=2xln⁡2f'(x) = 2^{x} \ln 2
Analyzing exponential growth with different bases.





Derivative Of Natural Logarithm



Formula:

ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}
The rate at which ln x changes is the reciprocal of x. As x increases, the rate decreases.
Natural logarithm of x
Reciprocal of x
If f(x)=ln⁡xf(x) = \ln x, then f′(x)=1xf'(x) = \frac{1}{x}
Solving time to reach a certain level in growth models.





Derivative Of Logarithm Base A



Formula:

ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}
For logarithms with any base a, the derivative is 1 over x times the natural log of the base.
Logarithm of x with base a
Natural logarithm of base a
If f(x)=log⁡2xf(x) = \log_2 x, then f′(x)=1xln⁡2f'(x) = \frac{1}{x \ln 2}
Working with logarithmic scales like decibels or pH.

Derivatives Of Inverse Trigonometric Functions






Derivative Of Inverse Sine



Formula:

ddx[arcsin⁡x]=11−x2\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}
The derivative of arcsin x shows how the inverse sine function changes with x. It grows faster as x approaches -1 or 1.
Inverse sine function
Square root of (1 minus x squared)
If f(x)=arcsin⁡xf(x) = \arcsin x, then f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}}
Calculating angles from ratios in trigonometry.





Derivative Of Inverse Cosine



Formula:

ddx[arccos⁡x]=−11−x2\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1 - x^2}}
Similar to arcsin x but negative, indicating the inverse cosine decreases as x increases.
Inverse cosine function
If f(x)=arccos⁡xf(x) = \arccos x, then f′(x)=−11−x2f'(x) = \frac{-1}{\sqrt{1 - x^2}}
Determining angles in physics problems.





Derivative Of Inverse Tangent



Formula:

ddx[arctan⁡x]=11+x2\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}
The derivative shows how the inverse tangent function changes smoothly over all real numbers.
Inverse tangent function
Sum of 1 and x squared
If f(x)=arctan⁡xf(x) = \arctan x, then f′(x)=11+x2f'(x) = \frac{1}{1 + x^2}
Signal processing, integrating to find angles.

Integrals






Indefinite Integral (Antiderivative)



Formula:

∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C
This tells us how to find the original function F(x) when we know its derivative f(x). It's like running differentiation in reverse to see where the function came from.
Function we're integrating (the derivative)
The original function (antiderivative)
Constant of integration (since constants vanish when differentiated)
If f(x)=2xf(x) = 2x, then ∫2x dx=x2+C\int 2x \, dx = x^2 + C
Recovering original quantities from rates, like position from velocity.





Definite Integral



Formula:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)
This calculates the net area under the curve f(x) from x = a to x = b. You find the antiderivative F(x), plug in the top limit b, subtract the value at the bottom limit a.
Lower and upper limits of integration
Function we're integrating
Antiderivative of f(x)
If f(x)=xf(x) = x, then ∫02x dx=[12x2]02=2\int_0^2 x \, dx = \left[ \frac{1}{2}x^2 \right]_0^2 = 2
Calculating total accumulated quantities, like distance traveled.





Improper Integral



Formula:

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx
When integrating over an infinite interval, replace infinity with a variable limit and take the limit as that variable approaches infinity.
Lower limit
Infinity (upper limit)
Compute ∫1∞1x2 dx=lim⁡b→∞(−1x∣1b)=1\int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left( -\frac{1}{x} \bigg|_1^b \right) = 1
Calculating probabilities in statistics, areas under curves extending to infinity.





Average Value Of A Function



Formula:

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx
This gives the average value of the function f(x) over the interval [a, b]. It's like finding the mean height of the curve over that interval.
Function we're averaging
Interval endpoints
Average value of f(x)
Find the average value of f(x)=x2f(x) = x^2 on [0, 2]: favg=12∫02x2 dx=12(83)=43f_{\text{avg}} = \frac{1}{2} \int_0^2 x^2 \, dx = \frac{1}{2} \left( \frac{8}{3} \right) = \frac{4}{3}
Physics (average velocity), economics (average cost).

Basic Integration Rules






Basic Integration Rule - Power Rule



Formula:

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n≠−1n \ne -1
To integrate x raised to a power, increase the exponent by one and divide by the new exponent. Don't forget the constant C!
Variable x raised to exponent n
Exponent (not equal to -1)
Constant of integration
If f(x)=x2f(x) = x^2, then ∫x2 dx=x33+C\int x^2 \, dx = \frac{x^{3}}{3} + C
Integrating polynomial functions.





Basic Integration Rule - Constant Multiple



Formula:

∫c⋅f(x) dx=c⋅∫f(x) dx\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx
If there's a constant multiplied by a function, you can pull it out of the integral and multiply it after integrating.
A constant number
Function to integrate
If f(x)=3xf(x) = 3x, then ∫3x dx=3⋅∫x dx=3⋅x22+C\int 3x \, dx = 3 \cdot \int x \, dx = 3 \cdot \frac{x^2}{2} + C
Simplifying integrals with constant coefficients.





Basic Integration Rule - Sum And Difference



Formula:

∫[f(x)±g(x)] dx=∫f(x) dx±∫g(x) dx\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx
You can split the integral of a sum or difference into separate integrals. Integrate each function individually, then add or subtract the results.
Functions being added or subtracted
If f(x)=x2+3xf(x) = x^2 + 3x, then ∫(x2+3x) dx=∫x2 dx+∫3x dx\int (x^2 + 3x) \, dx = \int x^2 \, dx + \int 3x \, dx
Breaking down complex integrals into simpler parts.

Integrals Of Trigonometric Functions






Integral Of Sine Function



Formula:

∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C
Integrating sin x gives us negative cos x. It's like asking, 'What function has a derivative of sin x?'
Sine function
Negative cosine function
Constant of integration
If f(x)=sin⁡xf(x) = \sin x, then ∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C
Solving problems involving oscillations, waves.





Integral Of Cosine Function



Formula:

∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C
The antiderivative of cos x is sin x. We're finding the function whose derivative is cos x.
Cosine function
Sine function
Constant of integration
If f(x)=cos⁡xf(x) = \cos x, then ∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C
Analyzing harmonic motion.

Integrals Of Exponential And Logarithmic Functions






Integral Of Exponential Function



Formula:

∫ex dx=ex+C\int e^{x} \, dx = e^{x} + C
The exponential function is its own antiderivative. Integrating e^x gives us e^x back.
Exponential function with base e
Constant of integration
If f(x)=exf(x) = e^{x}, then ∫ex dx=ex+C\int e^{x} \, dx = e^{x} + C
Modeling growth and decay processes.





Integral Of A^x



Formula:

∫ax dx=axln⁡a+C\int a^{x} \, dx = \frac{a^{x}}{\ln a} + C
When integrating an exponential function with base a, divide by the natural log of the base.
Exponential function with base a
Natural logarithm of base a
Constant of integration
If f(x)=2xf(x) = 2^{x}, then ∫2x dx=2xln⁡2+C\int 2^{x} \, dx = \frac{2^{x}}{\ln 2} + C
Calculations in finance, population models.





Integral Of Reciprocal Function



Formula:

∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln |x| + C
Integrating 1 over x gives the natural logarithm of the absolute value of x. It's like finding the area under the hyperbola y = 1/x.
Reciprocal function
Natural logarithm of the absolute value of x
Constant of integration
If f(x)=1xf(x) = \frac{1}{x}, then ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln |x| + C
Calculating time in decay processes.

Fundamental Theorem Of Calculus






Fundamental Theorem Of Calculus - Part 1



Formula:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)
This theorem bridges differentiation and integration. It tells us that the definite integral of a function over an interval equals the change in its antiderivative over that interval.
Function being integrated
Antiderivative of f(x)
Lower and upper limits
If f(x)=xf(x) = x, then F(x)=12x2F(x) = \frac{1}{2}x^2, so ∫13x dx=12(32)−12(12)=4\int_1^3 x \, dx = \frac{1}{2}(3^2) - \frac{1}{2}(1^2) = 4
Calculating net change, areas under curves.





Fundamental Theorem Of Calculus - Part 2



Formula:

ddx(∫axf(t) dt)=f(x)\frac{d}{dx} \left( \int_a^x f(t) \, dt \right) = f(x)
This part states that if you integrate a function and then differentiate the result, you get back the original function. Integration and differentiation are inverse processes.
Function inside the integral
Constant lower limit
Variable upper limit
If F(x)=∫0xcos⁡t dtF(x) = \int_0^x \cos t \, dt, then F′(x)=cos⁡xF'(x) = \cos x
Solving initial value problems, accumulating quantities.

Techniques Of Integration






Integration By Substitution (Reverse Chain Rule)



Formula:

If u=g(x)u = g(x), then ∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x)) g'(x) \, dx = \int f(u) \, du
When an integral contains a function and its derivative, you can substitute to simplify the integral. It's like undoing the chain rule from differentiation.
Substitution for simplifying
Composite function inside the integral
Derivative of inner function
Differential substitution
Integrate ∫2xcos⁡(x2) dx\int 2x \cos(x^2) \, dx by letting u=x2u = x^2, so du=2x dxdu = 2x \, dx
Simplifying integrals that are otherwise tough to solve.





Integration By Parts



Formula:

∫u dv=uv−∫v du\int u \, dv = u v - \int v \, du
This technique is useful when integrating a product of functions. Choose parts of the integrand as u and dv, differentiate u to get du, and integrate dv to get v.
Function to differentiate
Function to integrate
Derivative of u
Integral of dv
To integrate ∫xex dx\int x e^{x} \, dx, let u=xu = x (so du=dxdu = dx) and dv=exdxdv = e^{x} dx (so v=exv = e^{x})
Integrating products like x times an exponential or logarithm.





Partial Fractions Decomposition



Formula:

Decompose P(x)Q(x)\frac{P(x)}{Q(x)} into simpler fractions before integrating
If you have a rational function (polynomial over polynomial), break it into a sum of simpler fractions that are easier to integrate.
Polynomial in the numerator
Polynomial in the denominator
Integrate ∫2xx2−1 dx\int \frac{2x}{x^2 - 1} \, dx by decomposing into ∫(1x−1+1x+1)dx\int \left( \frac{1}{x - 1} + \frac{1}{x + 1} \right) dx
Integrating rational functions, solving differential equations.





Trigonometric Integrals



Formula:

Use identities to simplify ∫sin⁡nxcos⁡mx dx\int \sin^n x \cos^m x \, dx
When integrating powers of sine and cosine, use trigonometric identities to rewrite the integral into a more manageable form.
Like sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1, sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x
Integrate ∫sin⁡2x dx\int \sin^2 x \, dx by using sin⁡2x=1−cos⁡2x2\sin^2 x = \frac{1 - \cos 2x}{2}
Solving integrals involving trigonometric functions raised to powers.