Why is a constant of integration needed?

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

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

It would need division by zero; \int\frac1x\,dx gives a logarithm, covered later.

How do you integrate \frac{(x-1)^{2}}{\sqrt x}?

Expand and divide to x^{3/2}-2x^{1/2}+x^{-1/2}, then integrate each term.

How do you find the equation of a curve from f'(x)?

Integrate to get f(x)+c, then substitute a known point to find c.

Fundamental Theorem of Calculus (idea)?

Integration and differentiation are inverse processes.

How can you check an integral?

Differentiate your answer; you should get the original function.

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

Card 1 of 120 of 12 known

Question

$\int x^{n}\,dx$ for $n\neq-1$

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

$\int x^{n}\,dx$ for $n\neq-1$: $\frac{x^{n+1}}{n+1}+c$
Why is a constant of integration needed?: Constants differentiate to zero, so many functions share the same derivative.
Why is $n=-1$ excluded in the power rule?: It would need division by zero; $\int\frac1x\,dx$ gives a logarithm, covered later.
$\int\left(6x^{2}-4x+5\right)dx$: $2x^{3}-2x^{2}+5x+c$
$\int\sqrt x\,dx$: $\frac23x^{3/2}+c$
$\int x^{-3}\,dx$: $-\frac12x^{-2}+c$
$\int5\,dx$: $5x+c$
How do you integrate $\frac{(x-1)^{2}}{\sqrt x}$ ?: Expand and divide to $x^{3/2}-2x^{1/2}+x^{-1/2}$ , then integrate each term.
How do you find the equation of a curve from $f'(x)$ ?: Integrate to get $f(x)+c$ , then substitute a known point to find $c$ .
Fundamental Theorem of Calculus (idea)?: Integration and differentiation are inverse processes.
How can you check an integral?: Differentiate your answer; you should get the original function.
$f'(x)=2x-3$ and $f(1)=0$ . Find $f(x)$ .: $x^{2}-3x+2$

Exam questions on Integration as the reverse of differentiation

A curve has gradient function $\frac{dy}{dx}=6x^{2}-4x+5$ and passes through the point $(1,7)$ .
Find the $y$ -coordinate of the point on the curve where $x=2$ .2 marks
A curve $y=f(x)$ , for $x>0$ , has gradient function $f'(x)=\frac{(x-1)^{2}}{\sqrt x}$ and passes through the point $(1,2)$ .
Find an equation of the curve.2 marks
A curve $y=f(x)$ , for $x>0$ , has gradient function $f'(x)=3x^{2}-\frac{6}{x^{3}}$ and passes through the point $(1,2)$ .
Find $f(x)$ .3 marks

See the full worksheet

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).