State the rule for integrating x^n.

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

The derivative of a constant is zero, so the original constant is lost on differentiating.

Why does the power rule fail for n=-1?

The denominator n+1 would be zero.

Substitute the coordinates of a point on the curve into the integrated equation.

How do you integrate a quotient such as \frac{x^4+6x^2+9}{x^2}?

Split it into a sum of powers first: x^2+6+9x^{-2}.

How do you check an integral?

Differentiate the answer and compare with the original function.

Integration as the reverse of differentiationAQA A-Level Maths: Flashcards

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What is integration?: The reverse of differentiation.
State the rule for integrating $x^n$ .: $\int x^n\,dx=\frac{x^{n+1}}{n+1}+c$ for $n\neq-1$ .
Why is $+c$ needed?: The derivative of a constant is zero, so the original constant is lost on differentiating.
Why does the power rule fail for $n=-1$ ?: The denominator $n+1$ would be zero.
Find $\int x^{-2}\,dx$ .: $-\frac1x+c$
Find $\int\sqrt{x}\,dx$ .: $\frac23x^{\frac32}+c$
Find $\int6x^2\,dx$ .: $2x^3+c$
What does $\frac{d}{dx}\int f(x)\,dx$ equal?: $f(x)$
What does $\int f'(x)\,dx$ equal?: $f(x)+c$
How do you find $c$ ?: Substitute the coordinates of a point on the curve into the integrated equation.
How do you integrate a quotient such as $\frac{x^4+6x^2+9}{x^2}$ ?: Split it into a sum of powers first: $x^2+6+9x^{-2}$ .
How do you check an integral?: Differentiate the answer and compare with the original function.

Exam questions on Integration as the reverse of differentiation

The gradient of the curve $C$ is given by $\frac{dy}{dx}=6x^2-4x+5$ .
Find the value of $y$ on $C$ when $x=2$ .2 marks
The function $f$ satisfies $f'(x)=3\sqrt{x}+\frac{8}{x^2}$ for $x>0$ , and $f(4)=20$ .
Find the value of $f(1)$ .2 marks
The curve $C$ has gradient function $\frac{dy}{dx}=\frac{(x^2+3)^2}{x^2}$ for $x>0$ .
Find $\int\frac{dy}{dx}\,dx$ .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).