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

Which of the following is (x ) dx?

Choose u and (dv)/(dx) using LIATE: u=(x ), (dv)/(dx)=1 Substitute into the formula: (x ) dx=x (x )- 1 dx Select the correct result: (x ) dx=x ((x ) - 1) Answer: x ((x ) - 1)+c

Which of the following is x (x ) dx?

Write down the integral: x (x ) dx Choose u and (dv)/(dx) using LIATE: u=(x ), (dv)/(dx)=x Find (du)/(dx) and v: (du)/(dx)=(1)/(x), v=x^22 Substitute into the formula: x (x ) dx=x^2 (x )2- (x)/(2) dx Evaluate the remaining integral: (x)/(2) dx=x^24 Select the correct result: x (x ) dx=x^2 (2 (x ) - 1)4 Answer: x^2 (2 (x ) - 1)4+c

Using integration by parts, what is the exact value of _0^1 x e^x dx?

Write down the integral: x e^x dx Recall the integration by parts formula: u (dv)/(dx) dx=uv- v (du)/(dx) dx Choose u using LIATE: u=x Choose (dv)/(dx) (the other factor): (dv)/(dx)=e^x Differentiate u to find (du)/(dx): (du)/(dx)=1 Integrate (dv)/(dx) to find v: v=e^x Collect the four ingredients: u=x, (du)/(dx)=1, (dv)/(dx)=e^x, v=e^x Substitute into the formula: x e^x dx=x e^x- e^x dx Look at the remaining integral: e^x dx Evaluate the remaining integral: e^x dx=e^x Combine the two parts: x e^x-(e^x) Simplify: (x - 1) e^x Add the constant of integration: (x - 1) e^x+c Check by differentiating: (d)/(dx)((x - 1) e^x)=x e^x Select the correct formula: u dv=uv- v du Answer: 1

A-Level Integration by parts Practice Questions

Free A-Level Integration by parts practice questions with full step-by-step worked solutions. Covers integration by parts, LIATE, integrate ln x, loop integral. Practise exam-style problems and check your method.

integration by partsLIATEintegrate ln xloop integralarea under curve

A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int x e^{x}\,dx

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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

Which of the following is

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

Show worked solution

Worked solution

Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=\ln{\left(x \right)},\qquad \frac{dv}{dx}=1$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Substitute into the formula
$\int \ln{\left(x \right)}\,dx=x \ln{\left(x \right)}-\int 1\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
Select the correct result
$\int \ln{\left(x \right)}\,dx=x \left(\ln{\left(x \right)} - 1\right)$
Integration by parts gives this antiderivative (plus a constant).

Answer

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

Question 3

3 marksintermediate

Which of the following is

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

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Worked solution

Write down the integral
$\int x \ln{\left(x \right)}\,dx$
The integrand is a product of two different kinds of function, so integration by parts is appropriate.
Choose $u$ and $\dfrac{dv}{dx}$ using LIATE
$u=\ln{\left(x \right)},\qquad \frac{dv}{dx}=x$
Let $u$ be the factor that simplifies when differentiated; integrate the rest.
Find $\dfrac{du}{dx}$ and $v$
$\frac{du}{dx}=\frac{1}{x},\qquad v=\frac{x^{2}}{2}$
Differentiate $u$ and integrate $\dfrac{dv}{dx}$ .
Substitute into the formula
$\int x \ln{\left(x \right)}\,dx=\frac{x^{2} \ln{\left(x \right)}}{2}-\int \frac{x}{2}\,dx$
Replace $uv$ and $\int v\,\dfrac{du}{dx}\,dx$ using the pieces above.
Evaluate the remaining integral
$\int \frac{x}{2}\,dx=\frac{x^{2}}{4}$
Integrate again — a second application of integration by parts may be required.
Select the correct result
$\int x \ln{\left(x \right)}\,dx=\frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4}$
Integration by parts gives this antiderivative (plus a constant).

Answer

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

Question 4

5 markshard

Which of these integrals genuinely requires integration by parts?

Show worked solution

Worked solution

Write down the integral
$\int x e^{x}\,dx$
The integrand is a product of two different kinds of function, so integration by parts is appropriate.
Recall the integration by parts formula
$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$
This rewrites the integral of a product in terms of a simpler integral.
Choose $u$ using LIATE
$u=x$
LIATE (logs, inverse-trig, algebraic, trig, exponential) selects the factor to differentiate; it should become simpler.
Choose $\dfrac{dv}{dx}$ (the other factor)
$\frac{dv}{dx}=e^{x}$
The remaining factor is the part we integrate.
Differentiate $u$ to find $\dfrac{du}{dx}$
$\frac{du}{dx}=1$
Differentiate the chosen $u$ .
Integrate $\dfrac{dv}{dx}$ to find $v$
$v=e^{x}$
Integrate the other factor; no constant of integration is needed at this stage.
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.
Look at the remaining integral
$\int e^{x}\,dx$
This integral is simpler than the original.
Evaluate the remaining integral
$\int e^{x}\,dx=e^{x}$
Integrate again — a second application of integration by parts may be required.
Select the correct formula
$\int u\,dv=uv-\int v\,du$
This is the standard statement of integration by parts.

Answer

\int x e^{x}\,dx

Question 5

8 markschallenging

Using integration by parts, what is the exact value of

\int_{0}^{1} x e^{x}\,dx

Show worked solution

Worked solution

Write down the integral
$\int x e^{x}\,dx$
The integrand is a product of two different kinds of function, so integration by parts is appropriate.
Recall the integration by parts formula
$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$
This rewrites the integral of a product in terms of a simpler integral.
Choose $u$ using LIATE
$u=x$
LIATE (logs, inverse-trig, algebraic, trig, exponential) selects the factor to differentiate; it should become simpler.
Choose $\dfrac{dv}{dx}$ (the other factor)
$\frac{dv}{dx}=e^{x}$
The remaining factor is the part we integrate.
Differentiate $u$ to find $\dfrac{du}{dx}$
$\frac{du}{dx}=1$
Differentiate the chosen $u$ .
Integrate $\dfrac{dv}{dx}$ to find $v$
$v=e^{x}$
Integrate the other factor; no constant of integration is needed at this stage.
Collect the four ingredients
$u=x,\quad \frac{du}{dx}=1,\quad \frac{dv}{dx}=e^{x},\quad v=e^{x}$
Having every piece ready makes the substitution reliable.
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.
Look at the remaining integral
$\int e^{x}\,dx$
This integral is simpler than the original.
Evaluate the remaining integral
$\int e^{x}\,dx=e^{x}$
Integrate again — a second application of integration by parts may be required.
Combine the two parts
$x e^{x}-\left(e^{x}\right)$
Subtract the value of the remaining integral from $uv$ .
Simplify
$\left(x - 1\right) e^{x}$
Collect the terms into a single expression.
Add the constant of integration
$\left(x - 1\right) e^{x}+c$
An indefinite integral must include the arbitrary constant $+c$ .
Check by differentiating
$\frac{d}{dx}\left(\left(x - 1\right) e^{x}\right)=x e^{x}$
Differentiating the answer returns the original integrand, confirming the result.
Select the correct formula
$\int u\,dv=uv-\int v\,du$
This is the standard statement of integration by parts.

Answer

1

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