Find 2 x (x^2 + 1)^5 dx.

Choose the substitution and differentiate: u=x^2 + 1, du=2 x dx Rewrite in terms of u and integrate: u^5 du=u^66+c State the final answer: 2 x (x^2 + 1)^5 dx=(x^2 + 1)^66+c Answer: (x^2 + 1)^66+c

Choose the substitution and differentiate: u=x^2, du=2 x dx Rewrite in terms of u and integrate: e^u du=e^u+c State the final answer: 2 x e^x^2 dx=e^x^2+c Answer: e^x^2+c

Choose the substitution and differentiate: u=x^2, du=2 x dx Rewrite in terms of u and integrate: e^u2 du=e^u2+c State the final answer: x e^x^2 dx=e^x^22+c Answer: e^x^22+c

Choose the substitution and differentiate: u=x^2 + 1, du=2 x dx Rewrite in terms of u and integrate: (1)/(2 u) du=(u )2+c State the final answer: xx^2 + 1 dx=(x^2 + 1 )2+c Answer: (x^2 + 1 )2+c

Find 3 x^2x^3 + 1 dx.

Choose the substitution and differentiate: u=x^3 + 1, du=3 x^2 dx Rewrite in terms of u and integrate: (1)/(u) du=(u )+c State the final answer: 3 x^2x^3 + 1 dx=(x^3 + 1 )+c Answer: (x^3 + 1 )+c

Integration by substitution Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Integration by substitution questions. See exactly how to solve problems on substitution, reverse chain rule, exponential, logarithmic.

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A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Find

\int 2 x \left(x^{2} + 1\right)^{5}\,dx

Worked solution

Choose the substitution and differentiate
$u=x^{2} + 1,\quad du=2 x\,dx$
The derivative of the inner function is a factor of the integrand.
Rewrite in terms of u and integrate
$\int u^{5}\,du=\frac{u^{6}}{6}+c$
Integrate the standard expression in u.
State the final answer
$\int 2 x \left(x^{2} + 1\right)^{5}\,dx=\frac{\left(x^{2} + 1\right)^{6}}{6}+c$
Remember to include the constant of integration.

Answer

\frac{\left(x^{2} + 1\right)^{6}}{6}+c

Question 2

2 markseasy

Find

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

Worked solution

Choose the substitution and differentiate
$u=x^{2},\quad du=2 x\,dx$
The derivative of the inner function is a factor of the integrand.
Rewrite in terms of u and integrate
$\int e^{u}\,du=e^{u}+c$
Integrate the standard expression in u.
State the final answer
$\int 2 x e^{x^{2}}\,dx=e^{x^{2}}+c$
Remember to include the constant of integration.

Answer

e^{x^{2}}+c

Question 3

2 markseasy

Find

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

Worked solution

Choose the substitution and differentiate
$u=x^{2},\quad du=2 x\,dx$
The derivative of the inner function is a factor of the integrand.
Rewrite in terms of u and integrate
$\int \frac{e^{u}}{2}\,du=\frac{e^{u}}{2}+c$
Integrate the standard expression in u.
State the final answer
$\int x e^{x^{2}}\,dx=\frac{e^{x^{2}}}{2}+c$
Remember to include the constant of integration.

Answer

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

Question 4

2 markseasy

Find

\int \frac{x}{x^{2} + 1}\,dx

Worked solution

Choose the substitution and differentiate
$u=x^{2} + 1,\quad du=2 x\,dx$
The derivative of the inner function is a factor of the integrand.
Rewrite in terms of u and integrate
$\int \frac{1}{2 u}\,du=\frac{\ln{\left(u \right)}}{2}+c$
Integrate the standard expression in u.
State the final answer
$\int \frac{x}{x^{2} + 1}\,dx=\frac{\ln{\left(x^{2} + 1 \right)}}{2}+c$
Remember to include the constant of integration.

Answer

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

Question 5

2 markseasy

Find

\int \frac{3 x^{2}}{x^{3} + 1}\,dx

Worked solution

Choose the substitution and differentiate
$u=x^{3} + 1,\quad du=3 x^{2}\,dx$
The derivative of the inner function is a factor of the integrand.
Rewrite in terms of u and integrate
$\int \frac{1}{u}\,du=\ln{\left(u \right)}+c$
Integrate the standard expression in u.
State the final answer
$\int \frac{3 x^{2}}{x^{3} + 1}\,dx=\ln{\left(x^{3} + 1 \right)}+c$
Remember to include the constant of integration.

Answer

\ln{\left(x^{3} + 1 \right)}+c

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