Integration: Standard Results (Class 12)

This lesson is a reference list of the standard integration results for Class 12: the basic integrals, the rules for constants and sums, the integrals of some particular forms, and the integration by parts formula.

Basic standard integrals

For powers and the exponential and logarithmic families,

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1), \qquad \int \frac{1}{x} \, dx = \log|x| + C,$

$\int e^x \, dx = e^x + C, \qquad \int a^x \, dx = \frac{a^x}{\log a} + C, \qquad \int c \, dx = c\,x + C.$

Trigonometric integrals

The six basic trigonometric integrals are

$\int \sin x \, dx = -\cos x + C, \qquad \int \cos x \, dx = \sin x + C,$

$\int \sec^2 x \, dx = \tan x + C, \qquad \int \csc^2 x \, dx = -\cot x + C,$

$\int \sec x \tan x \, dx = \sec x + C, \qquad \int \csc x \cot x \, dx = -\csc x + C.$

The integrals of the four remaining ratios are

$\int \tan x \, dx = \log|\sec x| + C = -\log|\cos x| + C, \qquad \int \cot x \, dx = \log|\sin x| + C,$

$\int \sec x \, dx = \log|\sec x + \tan x| + C, \qquad \int \csc x \, dx = \log|\csc x - \cot x| + C.$

Inverse trigonometric forms

$\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1} x + C, \qquad \int \frac{1}{1 + x^2} \, dx = \tan^{-1} x + C,$

$\int \frac{1}{x\sqrt{x^2 - 1}} \, dx = \sec^{-1} x + C.$

Each has the matching negative inverse co-function as an equivalent answer, for example $-\cos^{-1} x$, $-\cot^{-1} x$, and $-\csc^{-1} x$.

Constant, sum, and zero rules

A constant factor comes straight outside the integral, and an integral distributes over a sum or difference:

$\int a\,f(x) \, dx = a \int f(x) \, dx, \qquad \int \big(f(x) \pm g(x)\big) \, dx = \int f(x) \, dx \pm \int g(x) \, dx.$

The integral of zero is just the constant of integration:

$\int 0 \, dx = C.$

Integrals of particular functions

These standard forms turn up repeatedly:

$\int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a} \log\left|\frac{x - a}{x + a}\right| + C,$

$\int \frac{1}{a^2 - x^2} \, dx = \frac{1}{2a} \log\left|\frac{a + x}{a - x}\right| + C,$

$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\frac{x}{a} + C.$

Square root forms

$\int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \log\left|x + \sqrt{x^2 - a^2}\right| + C,$

$\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\frac{x}{a} + C,$

$\int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \log\left|x + \sqrt{x^2 + a^2}\right| + C.$

For the square roots themselves,

$\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \log\left|x + \sqrt{x^2 - a^2}\right| + C,$

$\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \log\left|x + \sqrt{x^2 + a^2}\right| + C,$

$\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\frac{x}{a} + C.$

Integration by parts

For a product of two functions,

$\int u \, v \, dx = u \int v \, dx - \int \left(\frac{du}{dx} \int v \, dx\right) dx.$

In words, taking $u$ as the first function and $v$ as the second:

$\int (\text{first})(\text{second}) \, dx = (\text{first}) \int (\text{second}) \, dx - \int \left(\frac{d}{dx}(\text{first}) \int (\text{second}) \, dx\right) dx.$

Key takeaways

• Memorise the basic integrals of powers, exponentials, and all six trigonometric functions, each with its constant of integration.
• A constant pulls outside the integral and the integral splits over sums and differences.
• Know the particular forms with $x^2 \pm a^2$ and $a^2 - x^2$, both as fractions and under a square root.
• Integration by parts integrates a product: first times the integral of the second, minus the integral of the derivative of the first times the integral of the second.