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Integration Rules


Basic Integration Rules
Trigonometric Integrals
Inverse Trigonometric Integrals
Exponential and Logarithmic Integrals
Integration Techniques







Basic Integration Rules

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Constant Rule
∫c dx=cx+C\int c \, dx = cx + C
The integral of a constant is the constant times x plus the constant of integration
Constant Multiple Rule
∫c⋅f(x) dx=c∫f(x) dx\int c \cdot f(x) \, dx = c \int f(x) \, dx
Constants can be factored out of integrals
Power Rule
∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C (where n≠−1n \neq -1)
Add one to the exponent and divide by the new exponent
Sum and Difference Rule
∫[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
The integral of a sum/difference is the sum/difference of integrals
Special Power Rule
∫x−1 dx=∫1x dx=ln⁡∣x∣+C\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln|x| + C
The integral of one over x is the natural logarithm of absolute value of x

Trigonometric Integrals

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Sine Integral
∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C
The integral of sine is negative cosine
Cosine Integral
∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x) \, dx = \sin(x) + C
The integral of cosine is sine
Tangent Integral
∫tan⁡(x) dx=−ln⁡∣cos⁡(x)∣+C=ln⁡∣sec⁡(x)∣+C\int \tan(x) \, dx = -\ln|\cos(x)| + C = \ln|\sec(x)| + C
The integral of tangent is negative natural log of absolute cosine
Cotangent Integral
∫cot⁡(x) dx=ln⁡∣sin⁡(x)∣+C\int \cot(x) \, dx = \ln|\sin(x)| + C
The integral of cotangent is natural log of absolute sine
Secant Integral
∫sec⁡(x) dx=ln⁡∣sec⁡(x)+tan⁡(x)∣+C\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C
The integral of secant involves natural log of secant plus tangent
Cosecant Integral
∫csc⁡(x) dx=−ln⁡∣csc⁡(x)+cot⁡(x)∣+C\int \csc(x) \, dx = -\ln|\csc(x) + \cot(x)| + C
The integral of cosecant involves negative natural log of cosecant plus cotangent
Secant Squared Integral
∫sec⁡2(x) dx=tan⁡(x)+C\int \sec^2(x) \, dx = \tan(x) + C
The integral of secant squared is tangent
Cosecant Squared Integral
∫csc⁡2(x) dx=−cot⁡(x)+C\int \csc^2(x) \, dx = -\cot(x) + C
The integral of cosecant squared is negative cotangent
Secant Tangent Integral
∫sec⁡(x)tan⁡(x) dx=sec⁡(x)+C\int \sec(x)\tan(x) \, dx = \sec(x) + C
The integral of secant times tangent is secant
Cosecant Cotangent Integral
∫csc⁡(x)cot⁡(x) dx=−csc⁡(x)+C\int \csc(x)\cot(x) \, dx = -\csc(x) + C
The integral of cosecant times cotangent is negative cosecant

Inverse Trigonometric Integrals

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Arcsine Integral Form
∫11−x2 dx=arcsin⁡(x)+C\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C
Standard form that integrates to arcsine
Arctangent Integral Form
∫11+x2 dx=arctan⁡(x)+C\int \frac{1}{1+x^2} \, dx = \arctan(x) + C
Standard form that integrates to arctangent
Arcsecant Integral Form
∫1xx2−1,dx=arcsec∣x∣+C\int \frac{1}{x\sqrt{x^2-1}} ,dx =arcsec|x| + C
Standard form that integrates to arcsecant of absolute x
General Arcsine Form
∫1a2−x2 dx=arcsin⁡(xa)+C\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C
Generalized arcsine integral form with parameter a
General Arctangent Form
∫1a2+x2 dx=1aarctan⁡(xa)+C\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
Generalized arctangent integral form with parameter a

Exponential and Logarithmic Integrals

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Natural Exponential Integral
∫ex dx=ex+C\int e^x \, dx = e^x + C
The integral of e to the x is itself
General Exponential Integral
∫ax dx=axln⁡(a)+C\int a^x \, dx = \frac{a^x}{\ln(a)} + C (where a>0a > 0, a≠1a \neq 1)
The integral of exponential includes division by natural log of the base
Composite Natural Exponential
∫ef(x)f′(x) dx=ef(x)+C\int e^{f(x)} f'(x) \, dx = e^{f(x)} + C
Integration by substitution with natural exponential
Composite General Exponential
∫af(x)f′(x) dx=af(x)ln⁡(a)+C\int a^{f(x)} f'(x) \, dx = \frac{a^{f(x)}}{\ln(a)} + C
Integration by substitution with general exponential
Reciprocal Function Integral
∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C
The integral of derivative over function is natural log of absolute function

Integration Techniques

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Integration by Parts
∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du
Method for integrating products of functions
U-Substitution
∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x) \, dx = \int f(u) \, du where u=g(x)u = g(x)
Method for integrating composite functions using substitution
Fundamental Theorem of Calculus
∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a) where F′(x)=f(x)F'(x) = f(x)
Connects definite integrals with antiderivatives
Integration by Partial Fractions
∫P(x)Q(x) dx=∫(∑Ai(x−ri)ni)dx\int \frac{P(x)}{Q(x)} \, dx = \int \left(\sum \frac{A_i}{(x-r_i)^{n_i}}\right) dx
Method for integrating rational functions by decomposition