07Class 12 Maths

Integrals formulas

The standard integrals and methods for Class 12. Students who know this list well save a lot of time, so revise it often.

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Standard integrals

Power: $\displaystyle \int x^n\,dx = \frac{x^{n + 1}}{n + 1} + C, \quad n \ne -1$
Reciprocal and exponential: $\displaystyle \int\frac1x\,dx = \ln|x| + C, \quad \int e^x\,dx = e^x + C$
Sine and cosine: $\displaystyle \int\sin x\,dx = -\cos x + C, \quad \int\cos x\,dx = \sin x + C$
Squares of secant and cosecant: $\displaystyle \int\sec^2x\,dx = \tan x + C, \quad \int\csc^2x\,dx = -\cot x + C$
Tangent and cotangent: $\displaystyle \int\tan x\,dx = \ln|\sec x| + C, \quad \int\cot x\,dx = \ln|\sin x| + C$
Secant and cosecant: $\displaystyle \int\sec x\,dx = \ln|\sec x + \tan x| + C, \quad \int\csc x\,dx = \ln|\csc x - \cot x| + C$

Inverse tangent: $\displaystyle \int\frac{dx}{x^2 + a^2} = \frac1a\tan^{-1}\frac{x}{a} + C$
Inverse sine: $\displaystyle \int\frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C$
Log forms: $\displaystyle \int\frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| + C$
Root forms: $\displaystyle \int\frac{dx}{\sqrt{x^2 \pm a^2}} = \ln\left|x + \sqrt{x^2 \pm a^2}\right| + C$
Square root of a² − x²: $\displaystyle \int\sqrt{a^2 - x^2}\,dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$

Integration by parts: $\displaystyle \int u\,v\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx$
Choose u using the order ILATE.
A useful exponential form: $\displaystyle \int e^x[f(x) + f'(x)]\,dx = e^xf(x) + C$

Fundamental theorem: $\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$
King's property: $\displaystyle \int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$
Symmetric limits: $\displaystyle \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \ \text{if } f \text{ is even, and } 0 \text{ if odd}$

Dropping the + C in indefinite integrals.
Keeping the old limits after a substitution in a definite integral. Change them, or substitute back.
Using the power rule at n = −1. That case gives ln|x|.

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