Choose u and (dv)/(dx) using LIATE: u=x, (dv)/(dx)=e^x Substitute into the formula: x e^x dx=x e^x- e^x dx State the final answer: x e^x dx=(x - 1) e^x+c Answer: (x - 1) e^x+c

Choose u and (dv)/(dx) using LIATE: u=x, (dv)/(dx)=e^2 x Substitute into the formula: x e^2 x dx=x e^2 x2- e^2 x2 dx State the final answer: x e^2 x dx=(2 x - 1) e^2 x4+c Answer: (2 x - 1) e^2 x4+c

Choose u and (dv)/(dx) using LIATE: u=x, (dv)/(dx)=(x ) Substitute into the formula: x (x ) dx=- x (x )- - (x ) dx State the final answer: x (x ) dx=- x (x ) + (x )+c Answer: - x (x ) + (x )+c

Choose u and (dv)/(dx) using LIATE: u=x, (dv)/(dx)=(x ) Substitute into the formula: x (x ) dx=x (x )- (x ) dx State the final answer: x (x ) dx=x (x ) + (x )+c Answer: x (x ) + (x )+c

Choose u and (dv)/(dx) using LIATE: u=x, (dv)/(dx)=(2 x ) Substitute into the formula: x (2 x ) dx=- x (2 x )2- - (2 x )2 dx State the final answer: x (2 x ) dx=- x (2 x )2 + (2 x )4+c Answer: - x (2 x )2 + (2 x )4+c

Integration by parts Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Integration by parts questions. See exactly how to solve problems on integration by parts, LIATE, integrate ln x, loop integral.

integration by partsLIATEintegrate ln xloop integralarea under curve

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int x e^{x}\,dx

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=x,\qquad \frac{dv}{dx}=e^{x}$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int x e^{x}\,dx=x e^{x}-\int e^{x}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
State the final answer
$\int x e^{x}\,dx=\left(x - 1\right) e^{x}+c$
Combine the terms and include the constant of integration.

Answer

\left(x - 1\right) e^{x}+c

Question 2

2 markseasy

Find

\int x e^{2 x}\,dx

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=x,\qquad \frac{dv}{dx}=e^{2 x}$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int x e^{2 x}\,dx=\frac{x e^{2 x}}{2}-\int \frac{e^{2 x}}{2}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
State the final answer
$\int x e^{2 x}\,dx=\frac{\left(2 x - 1\right) e^{2 x}}{4}+c$
Combine the terms and include the constant of integration.

Answer

\frac{\left(2 x - 1\right) e^{2 x}}{4}+c

Question 3

2 markseasy

Find

\int x \sin{\left(x \right)}\,dx

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=x,\qquad \frac{dv}{dx}=\sin{\left(x \right)}$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int x \sin{\left(x \right)}\,dx=- x \cos{\left(x \right)}-\int - \cos{\left(x \right)}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
State the final answer
$\int x \sin{\left(x \right)}\,dx=- x \cos{\left(x \right)} + \sin{\left(x \right)}+c$
Combine the terms and include the constant of integration.

Answer

- x \cos{\left(x \right)} + \sin{\left(x \right)}+c

Question 4

2 markseasy

Find

\int x \cos{\left(x \right)}\,dx

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=x,\qquad \frac{dv}{dx}=\cos{\left(x \right)}$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int x \cos{\left(x \right)}\,dx=x \sin{\left(x \right)}-\int \sin{\left(x \right)}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
State the final answer
$\int x \cos{\left(x \right)}\,dx=x \sin{\left(x \right)} + \cos{\left(x \right)}+c$
Combine the terms and include the constant of integration.

Answer

x \sin{\left(x \right)} + \cos{\left(x \right)}+c

Question 5

2 markseasy

Find

\int x \sin{\left(2 x \right)}\,dx

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=x,\qquad \frac{dv}{dx}=\sin{\left(2 x \right)}$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int x \sin{\left(2 x \right)}\,dx=- \frac{x \cos{\left(2 x \right)}}{2}-\int - \frac{\cos{\left(2 x \right)}}{2}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
State the final answer
$\int x \sin{\left(2 x \right)}\,dx=- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}+c$
Combine the terms and include the constant of integration.

Answer

- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}+c

Unlock 65 more Integration by parts questions

Create a free account to work through every A-Level Integration by parts question with instant step-by-step worked solutions, progress tracking and interactive lessons.

Full worked solutions for every question
Interactive lessons and instant feedback
Track your mastery across every topic

Create a Free Account

No card required · Free forever

More Integration by parts practice

Hard Questions Practice Questions

Integration by parts Worked Solutions — A-Level Maths

Worked solution

Worked solution

Worked solution

Worked solution

Worked solution

Unlock 65 more Integration by parts questions

More Integration by parts practice

Related Pure Maths topics