Indefinite Integrals — Part 1: Standard forms and powers of sine and cosine: visual lesson, ISC Class 12 Mathematics

Lesson

Indefinite Integrals, Part 1

Differentiation tells you the slope from the curve. Integration asks the opposite question.

In this lesson you will

Describe integration as the reverse of differentiation and explain the constant of integration
Use the standard integrals of powers, the reciprocal, the exponential and the six trigonometric derivatives
Integrate functions of a linear expression ax + b
Integrate the squares, cubes and fourth powers of sine and cosine

Part 1 at a glance.

Concept

Each row is a derivative read from right to left.
Derivative known	Integral read off
$\dfrac{d}{dx}\,\dfrac{(ax + b)^{n + 1}}{a(n + 1)} = (ax + b)^n$	$\displaystyle\int (ax + b)^n\,dx = \dfrac{(ax + b)^{n + 1}}{a(n + 1)} + C$ , $n \ne -1$
$\dfrac{d}{dx}\log\|x\| = \dfrac1x$	$\displaystyle\int \dfrac{dx}{x} = \log\|x\| + C$
$\dfrac{d}{dx}e^x = e^x$	$\displaystyle\int e^x\,dx = e^x + C$
$\dfrac{d}{dx}(-\cos x) = \sin x$	$\displaystyle\int \sin x\,dx = -\cos x + C$
$\dfrac{d}{dx}\sin x = \cos x$	$\displaystyle\int \cos x\,dx = \sin x + C$
$\dfrac{d}{dx}\tan x = \sec^2 x$	$\displaystyle\int \sec^2 x\,dx = \tan x + C$
$\dfrac{d}{dx}(-\cot x) = \operatorname{cosec}^2 x$	$\displaystyle\int \operatorname{cosec}^2 x\,dx = -\cot x + C$

Concept

If $F'(x) = f(x)$ , then $\displaystyle\int f(x)\,dx = F(x) + C$ . Two antiderivatives of $f$ on an interval differ by a constant.

\int f(x)\,dx = F(x) + C

Every member of $x^2 + C$ has derivative $2x$: the curves are vertical shifts of one another.