Standard integrals

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What integration is

Integration (or anti-differentiation) is the reverse of differentiation: it finds a function from its derivative.
An indefinite integral has no limits and always includes the constant of integration $+ c$ , because differentiating removes any constant:

$\int f(x)\,dx = F(x) + c$

The reverse power rule

To integrate a power of $x$ , add 1 to the power and divide by the new power:

$\int x^n\,dx = \frac{x^{\,n+1}}{n + 1} + c \qquad (n \neq -1)$

Integrate term by term for a whole expression.

The other standard integrals

The special case $n = -1$ and the trig/exponential integrals must be learned:

$\int \frac{1}{x}\,dx = \ln|x| + c \qquad \int e^x\,dx = e^x + c$

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

Worked ExampleIntegrating a polynomial

Find $\displaystyle\int (6x^2 - 4x + 1)\,dx$ .

Step 1 — Integrate each term with the reverse power rule

$6x^2 \to \dfrac{6x^3}{3} = 2x^3$
$-4x \to \dfrac{-4x^2}{2} = -2x^2$
$1 \to x$ (since $1 = x^0$ , adding 1 to the power gives $x^1$ )

Step 2 — Add the constant of integration

$\int (6x^2 - 4x + 1)\,dx = 2x^3 - 2x^2 + x + c$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find $\displaystyle\int 4x^3\,dx$ .

Merit

Find $\displaystyle\int \left(e^x + \cos x\right)\,dx$ .

Excellence

Find $\displaystyle\int \left(\dfrac{3}{x} + \sqrt{x}\right)\,dx$ .