Write down the integral: x^3 dx Recall the power rule for integration: x^n dx=x^n+1n+1+c(n≠-1) Integrate term by term: (1)/(4)x^4 Add the constant of integration: (1)/(4)x^4+c Answer: (1)/(4)x^4+c

Write down the integral: x^5 dx Recall the power rule for integration: x^n dx=x^n+1n+1+c(n≠-1) Integrate term by term: (1)/(6)x^6 Add the constant of integration: (1)/(6)x^6+c Answer: (1)/(6)x^6+c

Write down the integral: x^2 dx Recall the power rule for integration: x^n dx=x^n+1n+1+c(n≠-1) Integrate term by term: (1)/(3)x^3 Add the constant of integration: (1)/(3)x^3+c Answer: (1)/(3)x^3+c

Write down the integral: 4x^3 dx Recall the power rule for integration: x^n dx=x^n+1n+1+c(n≠-1) Integrate term by term: x^4 Add the constant of integration: x^4+c Answer: x^4+c

Write down the integral: 6x^2 dx Recall the power rule for integration: x^n dx=x^n+1n+1+c(n≠-1) Integrate term by term: 2x^3 Add the constant of integration: 2x^3+c Answer: 2x^3+c

Indefinite integration Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Indefinite integration questions. See exactly how to solve problems on power rule, integrate x^n, coefficient, sum rule.

power ruleintegrate x^ncoefficientsum ruleconstantlinear

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int x^{3}\,dx

Worked solution

Write down the integral
$\int x^{3}\,dx$
Integration is the reverse of differentiation, so we look for a function whose derivative is the integrand.
Recall the power rule for integration
$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+c\quad(n\neq-1)$
For every power except $-1$ we add one to the power and divide by the new power.
Integrate term by term
$\frac{1}{4}x^{4}$
Apply the power rule to each term; any number multiplying $x$ simply carries through unchanged.
Add the constant of integration
$\frac{1}{4}x^{4}+c$
Differentiating a constant gives zero, so when we reverse the process we must include an unknown constant $c$ .

Answer

\frac{1}{4}x^{4}+c

Question 2

2 markseasy

Find

\int x^{5}\,dx

Worked solution

Write down the integral
$\int x^{5}\,dx$
Integration is the reverse of differentiation, so we look for a function whose derivative is the integrand.
Recall the power rule for integration
$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+c\quad(n\neq-1)$
For every power except $-1$ we add one to the power and divide by the new power.
Integrate term by term
$\frac{1}{6}x^{6}$
Apply the power rule to each term; any number multiplying $x$ simply carries through unchanged.
Add the constant of integration
$\frac{1}{6}x^{6}+c$
Differentiating a constant gives zero, so when we reverse the process we must include an unknown constant $c$ .

Answer

\frac{1}{6}x^{6}+c

Question 3

2 markseasy

Find

\int x^{2}\,dx

Worked solution

Write down the integral
$\int x^{2}\,dx$
Integration is the reverse of differentiation, so we look for a function whose derivative is the integrand.
Recall the power rule for integration
$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+c\quad(n\neq-1)$
For every power except $-1$ we add one to the power and divide by the new power.
Integrate term by term
$\frac{1}{3}x^{3}$
Apply the power rule to each term; any number multiplying $x$ simply carries through unchanged.
Add the constant of integration
$\frac{1}{3}x^{3}+c$
Differentiating a constant gives zero, so when we reverse the process we must include an unknown constant $c$ .

Answer

\frac{1}{3}x^{3}+c

Question 4

2 markseasy

Find

\int 4x^{3}\,dx

Worked solution

Write down the integral
$\int 4x^{3}\,dx$
Integration is the reverse of differentiation, so we look for a function whose derivative is the integrand.
Recall the power rule for integration
$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+c\quad(n\neq-1)$
For every power except $-1$ we add one to the power and divide by the new power.
Integrate term by term
$x^{4}$
Apply the power rule to each term; any number multiplying $x$ simply carries through unchanged.
Add the constant of integration
$x^{4}+c$
Differentiating a constant gives zero, so when we reverse the process we must include an unknown constant $c$ .

Answer

x^{4}+c

Question 5

2 markseasy

Find

\int 6x^{2}\,dx

Worked solution

Write down the integral
$\int 6x^{2}\,dx$
Integration is the reverse of differentiation, so we look for a function whose derivative is the integrand.
Recall the power rule for integration
$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+c\quad(n\neq-1)$
For every power except $-1$ we add one to the power and divide by the new power.
Integrate term by term
$2x^{3}$
Apply the power rule to each term; any number multiplying $x$ simply carries through unchanged.
Add the constant of integration
$2x^{3}+c$
Differentiating a constant gives zero, so when we reverse the process we must include an unknown constant $c$ .

Answer

2x^{3}+c

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