Sec 4 A-Math · Integration

Integration as the reverse of differentiation questions with answers

Practice questions on integration as the reverse of differentiation, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Given that $\dfrac{d}{dx}\left((x^{2} + 2)^{3}\right) = 6x(x^{2} + 2)^{2}$ , find $\int 3x(x^{2} + 2)^{2} \, dx$ .

Step 1 of 3

Integration reverses differentiation: integrate both sides of the given result

\int 6x(x^{2} + 2)^{2} \, dx = (x^{2} + 2)^{3} + c

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Questions

Question 1A multiple of the derivative
Given that $\dfrac{d}{dx}\left((x^{2} + 1)^{3}\right) = 6x(x^{2} + 1)^{2}$ , find $\int x(x^{2} + 1)^{2} \, dx$ .
Show the worked solution
1. Step 1: Integration reverses differentiation: integrate both sides of the given result $\int 6x(x^{2} + 1)^{2} \, dx = (x^{2} + 1)^{3} + c$
2. Step 2: The integrand you want is a multiple of the given derivative: find the multiple $x(x^{2} + 1)^{2} = \dfrac{1}{6} \times \left(6x(x^{2} + 1)^{2}\right)$
3. Step 3: So multiply the result by the same number $\int x(x^{2} + 1)^{2} \, dx = \dfrac{1}{6}(x^{2} + 1)^{3} + c$
Question 2A multiple of the derivative
Given that $\dfrac{d}{dx}\left(\ln(x^{2} + 1)\right) = \dfrac{2x}{x^{2} + 1}$ , find $\int \dfrac{x}{x^{2} + 1} \, dx$ .
Show the worked solution
1. Step 1: Integration reverses differentiation: integrate both sides of the given result $\int \dfrac{2x}{x^{2} + 1} \, dx = \ln(x^{2} + 1) + c$
2. Step 2: The integrand you want is a multiple of the given derivative: find the multiple $\dfrac{x}{x^{2} + 1} = \dfrac{1}{2} \times \left(\dfrac{2x}{x^{2} + 1}\right)$
3. Step 3: So multiply the result by the same number $\int \dfrac{x}{x^{2} + 1} \, dx = \dfrac{1}{2}\ln(x^{2} + 1) + c$
Question 3Rearrange first
Given that $\dfrac{d}{dx}\left(x^{2}\ln x\right) = 2x\ln x + x$ , find $\int x\ln x \, dx$ .
Show the worked solution
1. Step 1: Integrate both sides of the given result: integration reverses differentiation $x^{2}\ln x = \int 2x\ln x \, dx + \int x \, dx$
2. Step 2: Integrate the other part, x $\int x \, dx = \dfrac{x^{2}}{2}$
3. Step 3: Rearrange to make the integral you need the subject $\int 2x\ln x \, dx = x^{2}\ln x - \dfrac{x^{2}}{2}$
4. Step 4: Scale to the integral asked for $\int x\ln x \, dx = \dfrac{1}{2}x^{2}\ln x - \dfrac{1}{4}x^{2} + c$
Question 4Rearrange first
Given that $\dfrac{d}{dx}\left(x\sin x\right) = x\cos x + \sin x$ , find $\int x\cos x \, dx$ .
Show the worked solution
1. Step 1: Integrate both sides of the given result: integration reverses differentiation $x\sin x = \int x\cos x \, dx + \int \sin x \, dx$
2. Step 2: Integrate the other part, the sine $\int \sin x \, dx = -\dfrac{\cos x}{1}$
3. Step 3: Rearrange to make the integral you need the subject $\int x\cos x \, dx = x\sin x - \left(-\dfrac{\cos x}{1}\right)$
4. Step 4: Scale to the integral asked for $\int x\cos x \, dx = x\sin x + \cos x + c$

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Worked example

Given that

\dfrac{d}{dx}\left((x^{2} + 2)^{3}\right) = 6x(x^{2} + 2)^{2}

, find

\int 3x(x^{2} + 2)^{2} \, dx

Step 1 of 3

Integration reverses differentiation: integrate both sides of the given result

\int 6x(x^{2} + 2)^{2} \, dx = (x^{2} + 2)^{3} + c

Questions

Question 1A multiple of the derivative

Given that $\dfrac{d}{dx}\left((x^{2} + 1)^{3}\right) = 6x(x^{2} + 1)^{2}$ , find $\int x(x^{2} + 1)^{2} \, dx$ .

Show the worked solution

Step 1: Integration reverses differentiation: integrate both sides of the given result $\int 6x(x^{2} + 1)^{2} \, dx = (x^{2} + 1)^{3} + c$
Step 2: The integrand you want is a multiple of the given derivative: find the multiple $x(x^{2} + 1)^{2} = \dfrac{1}{6} \times \left(6x(x^{2} + 1)^{2}\right)$
Step 3: So multiply the result by the same number $\int x(x^{2} + 1)^{2} \, dx = \dfrac{1}{6}(x^{2} + 1)^{3} + c$

Question 2A multiple of the derivative

Given that $\dfrac{d}{dx}\left(\ln(x^{2} + 1)\right) = \dfrac{2x}{x^{2} + 1}$ , find $\int \dfrac{x}{x^{2} + 1} \, dx$ .

Show the worked solution

Step 1: Integration reverses differentiation: integrate both sides of the given result $\int \dfrac{2x}{x^{2} + 1} \, dx = \ln(x^{2} + 1) + c$
Step 2: The integrand you want is a multiple of the given derivative: find the multiple $\dfrac{x}{x^{2} + 1} = \dfrac{1}{2} \times \left(\dfrac{2x}{x^{2} + 1}\right)$
Step 3: So multiply the result by the same number $\int \dfrac{x}{x^{2} + 1} \, dx = \dfrac{1}{2}\ln(x^{2} + 1) + c$

Question 3Rearrange first

Given that $\dfrac{d}{dx}\left(x^{2}\ln x\right) = 2x\ln x + x$ , find $\int x\ln x \, dx$ .

Show the worked solution

Step 1: Integrate both sides of the given result: integration reverses differentiation $x^{2}\ln x = \int 2x\ln x \, dx + \int x \, dx$
Step 2: Integrate the other part, x $\int x \, dx = \dfrac{x^{2}}{2}$
Step 3: Rearrange to make the integral you need the subject $\int 2x\ln x \, dx = x^{2}\ln x - \dfrac{x^{2}}{2}$
Step 4: Scale to the integral asked for $\int x\ln x \, dx = \dfrac{1}{2}x^{2}\ln x - \dfrac{1}{4}x^{2} + c$

Question 4Rearrange first

Given that $\dfrac{d}{dx}\left(x\sin x\right) = x\cos x + \sin x$ , find $\int x\cos x \, dx$ .

Show the worked solution

Step 1: Integrate both sides of the given result: integration reverses differentiation $x\sin x = \int x\cos x \, dx + \int \sin x \, dx$
Step 2: Integrate the other part, the sine $\int \sin x \, dx = -\dfrac{\cos x}{1}$
Step 3: Rearrange to make the integral you need the subject $\int x\cos x \, dx = x\sin x - \left(-\dfrac{\cos x}{1}\right)$
Step 4: Scale to the integral asked for $\int x\cos x \, dx = x\sin x + \cos x + c$