Integration · Guided Practice

Integration

26 questions with answers and video solutions. Try each one before revealing the answer.

1
Find ∫(4x−3x2+1) dx\displaystyle \int\left(4\sqrt{x}-\frac{3}{x^2}+1\right)\,dx
Video solution coming soon

No hints on this one — its full written solution is with the answer.

2
Find ∫2x4 − 5x3 dx, x≠0\displaystyle \int\frac{2x^4\,-\,5}{x^3}\,dx,\ x\neq 0
Video solution coming soon

No hints on this one — its full written solution is with the answer.

3
Evaluate ∫1413x2 dx\displaystyle \int^{4}_{1}\frac{1}{3x^2}\,dx
Video solution coming soon

No hints on this one — its full written solution is with the answer.

4
Evaluate ∫23 (4x3−2x) dx\displaystyle \int^{\sqrt{3}}_{\sqrt{2}}\:\left(4x^3-2x\right)\,dx
Video solution coming soon

No hints on this one — its full written solution is with the answer.

5
Find ∫(2x+3)5 dx\displaystyle \int(2x+3)^5\,dx
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No hints on this one — its full written solution is with the answer.

6
Find ∫4(9 − x)6 dx, x≠9\displaystyle \int\frac{4}{(9\,-\,x)^6}\,dx,\ x\neq 9
Video solution coming soon

No hints on this one — its full written solution is with the answer.

7
Find ∫3sin⁡(2x−π6) dx\displaystyle \int 3\sin\left(2x-\frac{\pi}{6}\right)\,dx
Video solution coming soon

No hints on this one — its full written solution is with the answer.

8
Evaluate ∫0π65cos⁡(3x+π4) dx\displaystyle \int^{\frac{\pi}{6}}_{0}5\cos\left(3x+\frac{\pi}{4}\right)\,dx
Video solution coming soon

No hints on this one — its full written solution is with the answer.

9
For a function f\displaystyle f, defined on a suitable domain, it is known that: f′(x)=3x − 2x\displaystyle f'(x)=\frac{3x\,-\,2}{\sqrt{x}} and f(9)=45\displaystyle f(9)=45.
Express f(x)\displaystyle f(x) in terms of x\displaystyle x.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

10
A curve is such that dydx=6x2+1x2\displaystyle \frac{dy}{dx} =6x^2+\frac{1}{x^2}. The curve passes through the point (−1, 3)\displaystyle (-1,\,3).
Express y\displaystyle y in terms of x\displaystyle x.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

11
Given that f(x)=3x2−12\displaystyle f(x)=3x^2-12, find the area enclosed by the graph of y=f(x)\displaystyle y=f(x) and the x\displaystyle x-axis.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

12
Find the area enclosed by the parabola y=−x2+2x\displaystyle y=-x^2+2x and the straight line y=3x−12\displaystyle y=3x-12.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

13
2017 P1 Q13
4 Marks
Find
∫1(5−4x)12dx\displaystyle \int\frac{1}{(5-4x)^{\frac{1}{2}}}dx, x<54.\displaystyle x \lt \frac{5}{4}.
14
2018 P1 Q10
4 Marks

Given that
dydx=6x2−3x+4\displaystyle \frac{dy}{dx}=6x^{2}-3x+4
and y=14\displaystyle y=14 when x=2\displaystyle x=2, express y\displaystyle y in terms of x\displaystyle x.

15
2019 P1 Q11
4 Marks
Evaluate
∫0π9cos⁡(3x−π6)dx.\displaystyle \int_{0}^{\frac{\pi}{9}}\cos(3x-\frac{\pi}{6})dx.
16
2019 P2 Q2
4 Marks
Find
∫(6x−4x−3+5)dx.\displaystyle \int(6\sqrt{x}-4x^{-3}+5)dx.
17
2022 P1 Q6
4 Marks
Evaluate
∫−52(10−3x)−12dx\displaystyle \int_{-5}^{2}(10-3x)^{-\frac{1}{2}}dx
18
2023 P1 Q6
4 Marks
Find
∫(2x5−6x)dx,x≥0\displaystyle \int(2x^{5}-6\sqrt{x})dx, x\ge0
19
2024 P2 Q5
3 Marks

Evaluate
∫0π7sin⁡5xdx\displaystyle \int_{0}^{\frac{\pi}{7}}\sin 5x dx

20
2024 P2 Q7
5 Marks

The diagram shows the curve with equation y=x3−6x2+11x\displaystyle y=x^{3}-6x^{2}+11x intersecting the curve with equation y=6+4x−2x2\displaystyle y=6+4x-2x^{2} at x=2\displaystyle x=2

Intersection of cubic and quadratic curves

Calculate the shaded area.

21
2025 P1 Q12
4 Marks

Given that:
dydx=6cos⁡x+8sin⁡2x\displaystyle \frac{dy}{dx}=6 \cos x+8 \sin 2x
and y=4\displaystyle y=4 when x=π6\displaystyle x=\frac{\pi}{6}
express y in terms of x.

22
2025 P2 Q3
4 Marks

The diagram shows the graph of y=x2−2x+3.\displaystyle y=x^{2}-2x+3.

Graph of y=x^2-2x+3 with shaded area between x=2 and x=4

Calculate the shaded area.

23
2025 P2 Q7
2 Marks
Find
∫(3x+2)7dx\displaystyle \int(3x+2)^{7}dx.
24
2026 P1 Q2
4 Marks
Find
∫(15x23+7x2)dx,\displaystyle \int\left(15x^{\frac{2}{3}}+\frac{7}{x^{2}}\right)dx, x>0.\displaystyle x \gt 0.
25
2026 P1 Q10
4 Marks
Find
∫π6π36sin⁡(3x−π2)dx.\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}6\sin\left(3x-\frac{\pi}{2}\right)dx.
26
2026 P2 Q7
4 Marks

The diagram shows the curve with equation y=x2−x−2.\displaystyle y=x^{2}-x-2.

Curve y = x squared minus x minus 2, with the region between the curve and the x-axis shaded

Calculate the shaded area.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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