Write down the integral: x^3 dx Apply the standard result: x^3 dx = (1)/(4)x^4 Add the constant of integration: (1)/(4)x^4 + c Answer: (1)/(4)x^4 + c

Write down the integral: 6x^2 dx Apply the standard result: 6x^2 dx = 2x^3 Add the constant of integration: 2x^3 + c Answer: 2x^3 + c

Write down the integral: 4x dx Apply the standard result: 4x dx = 2x^2 Add the constant of integration: 2x^2 + c Answer: 2x^2 + c

Write down the integral: e^2 x dx Apply the standard result: e^2 x dx = (1)/(2)e^2 x Add the constant of integration: (1)/(2)e^2 x + c Answer: (1)/(2)e^2 x + c

Write down the integral: 6e^3 x dx Apply the standard result: 6e^3 x dx = 2e^3 x Add the constant of integration: 2e^3 x + c Answer: 2e^3 x + c

Standard integrals Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Standard integrals questions. See exactly how to solve problems on standard integrals, power rule, exponential, trigonometric.

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A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int x^{3}\,dx

Worked solution

Write down the integral
$\int x^{3}\,dx$
We need a function whose derivative is the integrand.
Apply the standard result
$\int x^{3}\,dx = \frac{1}{4}x^{4}$
Use the formula-book result for this type of term.
Add the constant of integration
$\frac{1}{4}x^{4} + c$
An indefinite integral always includes an arbitrary constant c.

Answer

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

Question 2

2 markseasy

Find

\int 6x^{2}\,dx

Worked solution

Write down the integral
$\int 6x^{2}\,dx$
We need a function whose derivative is the integrand.
Apply the standard result
$\int 6x^{2}\,dx = 2x^{3}$
Use the formula-book result for this type of term.
Add the constant of integration
$2x^{3} + c$
An indefinite integral always includes an arbitrary constant c.

Answer

2x^{3} + c

Question 3

2 markseasy

Find

\int 4x\,dx

Worked solution

Write down the integral
$\int 4x\,dx$
We need a function whose derivative is the integrand.
Apply the standard result
$\int 4x\,dx = 2x^{2}$
Use the formula-book result for this type of term.
Add the constant of integration
$2x^{2} + c$
An indefinite integral always includes an arbitrary constant c.

Answer

2x^{2} + c

Question 4

2 markseasy

Find

\int e^{2 x}\,dx

Worked solution

Write down the integral
$\int e^{2 x}\,dx$
We need a function whose derivative is the integrand.
Apply the standard result
$\int e^{2 x}\,dx = \frac{1}{2}e^{2 x}$
Use the formula-book result for this type of term.
Add the constant of integration
$\frac{1}{2}e^{2 x} + c$
An indefinite integral always includes an arbitrary constant c.

Answer

\frac{1}{2}e^{2 x} + c

Question 5

2 markseasy

Find

\int 6e^{3 x}\,dx

Worked solution

Write down the integral
$\int 6e^{3 x}\,dx$
We need a function whose derivative is the integrand.
Apply the standard result
$\int 6e^{3 x}\,dx = 2e^{3 x}$
Use the formula-book result for this type of term.
Add the constant of integration
$2e^{3 x} + c$
An indefinite integral always includes an arbitrary constant c.

Answer

2e^{3 x} + c

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