State the rule for \int x^n\,dx.

\frac{x^{n+1}}{n+1}+c, for n\neq-1.

Why is +c needed in an indefinite integral?

Constants differentiate to 0, so many functions share the same derivative.

How can you check an indefinite integral?

Differentiate it; you should get the integrand.

Why is n=-1 excluded from the power rule?

The rule would divide by n+1=0.

First step for \int\frac{x^3-2x}{\sqrt{x}}\,dx?

Divide each term: x^{\frac52}-2x^{\frac12}.

What is the first step for \int(2x+1)^2x^{-\frac12}\,dx?

Expand (2x+1)^2 and multiply each term by x^{-\frac12}.

Is \int6x^2\,dx=6x^3+c?

No. It is 2x^3+c: divide by the new power.

Indefinite integrationEdexcel International A Level Maths: Flashcards

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State the rule for $\int x^n\,\mathrm{d}x$.

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State the rule for $\int x^n\,\mathrm{d}x$ .: $\frac{x^{n+1}}{n+1}+c$ , for $n\neq-1$ .
Why is $+c$ needed in an indefinite integral?: Constants differentiate to $0$ , so many functions share the same derivative.
$\int k\,\mathrm{d}x$ for a constant $k$ ?: $kx+c$
$\int(6x^2-4x+5)\,\mathrm{d}x$ ?: $2x^3-2x^2+5x+c$
$\int x^{-3}\,\mathrm{d}x$ ?: $-\frac12x^{-2}+c$
$\int\sqrt{x}\,\mathrm{d}x$ ?: $\frac23x^{\frac32}+c$
$\int\frac{1}{\sqrt{x}}\,\mathrm{d}x$ ?: $2x^{\frac12}+c$
$\int\frac{3}{x^4}\,\mathrm{d}x$ ?: $-x^{-3}+c$
How can you check an indefinite integral?: Differentiate it; you should get the integrand.
Why is $n=-1$ excluded from the power rule?: The rule would divide by $n+1=0$ .
First step for $\int\frac{x^3-2x}{\sqrt{x}}\,\mathrm{d}x$ ?: Divide each term: $x^{\frac52}-2x^{\frac12}$ .
What is the first step for $\int(2x+1)^2x^{-\frac12}\,\mathrm{d}x$ ?: Expand $(2x+1)^2$ and multiply each term by $x^{-\frac12}$ .
Is $\int6x^2\,\mathrm{d}x=6x^3+c$ ?: No. It is $2x^3+c$ : divide by the new power.

Exam questions on Indefinite integration

Let $g(x)=6x^2-4x+5$ .
Find $\int\left(g(x)+\frac{4}{x^3}\right)\mathrm{d}x$ .2 marks
A function is defined by $f(x)=4\sqrt{x}-\dfrac{6}{x^3}$ for $x>0$ .
Find $\int x^3f(x)\,\mathrm{d}x$ .2 marks
The function $f$ is defined by $f(x)=\dfrac{(2x+1)^2}{\sqrt{x}}$ for $x>0$ .
Show that $f(x)=4x^{\frac32}+4x^{\frac12}+x^{-\frac12}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).