• A Level Maths

Demo

• A Level
• Integration (By ...)

1. Use the substitution $u = \sqrt{x+1}$ to find $$\int_0^1 \dfrac{x}{\sqrt{x+1}} \ \mathrm{d}x$$

$\dfrac{4-2\sqrt{2}}{3}$

2. Use the substitution $u = \sqrt{2x+1}$ to find $$\int_0^1 x\sqrt{2x+1} \ \mathrm{d}x$$

$\dfrac{6\sqrt{3}+1}{15}$

3. Use integration by parts to find $$\int x\sin x \ \mathrm{d}x$$

$\sin x - x\cos x + c$

4. Use integration by parts to find $$\int x\cos 3x \ \mathrm{d}x$$

$\dfrac{1}{9}(3x\sin(3x)+\cos(3x))$

5. Find the area bound by the $x$ axis, the lines $x = 4$ and $x = 11$, and the curve with parametric equations $$x = t^3+3 \quad y = \dfrac{2}{3t}$$

$3$

6. Find the finite area bound by the curve with parametric equations $$x = 3t+\sin t \quad y = 2\sin t; \quad 0 \leqslant t \leqslant \pi$$ and the $x$ axis.

$12$

7. Find the area bound by the $x$ axis, the lines $x = 7$ and $x = 15$, and the curve with parametric equations $$x = 4t-1 \quad y = 16t^{-2}$$

$16$

8. The curve with parametric equations $$x = 4 - t^2 \quad y = t^2+3t$$ crosses both the $x$ axis and the $y$ axis in the first quadrant. Find the finite area bound by the curve and the positive coordinate axes.

$24$

9. Use a suitable substitution to find $$\int \sqrt{x}\sqrt{x\sqrt{x}+1} \ \mathrm{d}x$$

$\dfrac{4}{9}(x^{\frac{3}{2}}+1)^{\frac{3}{2}}+c$

10. Use a suitable substitution to find $$\int x^3\sqrt{x^2+1} \ \mathrm{d}x$$

$\dfrac{(3x^2-2)(x^2+1)^{\frac{3}{2}}}{15}+c$

11. Use a suitable substitution to find $$\int (x^2+1)\sqrt{x-2} \ \mathrm{d}x$$

$\dfrac{2(x-2)^{\frac{3}{2}}}{105}\left(15x^2 + 24x + 57 \right)+c$

12. Use integration by parts to find $$\int x\mathrm{e}^{x} \ \mathrm{d}x$$

$\mathrm{e}^x(x-1)+c$

13. Use a suitable substitution to find $$\int \dfrac{x^2+2x}{x^2+2x+1} \ \mathrm{d}x$$

$x+\dfrac{1}{x+1}+c$

14. Use integration by parts to find $$\int 3x\mathrm{e}^{-x} \ \mathrm{d}x$$

$-3\mathrm{e}^{-x}(x+1) + c$

15. Use integration by parts to find $$\int 2x^2\mathrm{e}^x \ \mathrm{d}x$$

$2\mathrm{e}^x(x^2-2x+2)+c$

16. Find $$\int \dfrac{1}{x^2+6x+9} \ \mathrm{d}x$$

$-\dfrac{1}{x+3}+c$

17. Find $$\int x\tan(x^2)\sec(x^2) \ \mathrm{d}x$$

$\dfrac{1}{2}\sec(x^2)+c$

18. Find $$\int \dfrac{\ln x}{x^2} \ \mathrm{d}x$$

$-\dfrac{1}{x}(\ln x + 1) + c$

19. Find $$\int 3x^3(x^2+4)^5 \ \mathrm{d}x$$

$\dfrac{(3x^2-2)(x^2+4)^6}{14}+c$

20. Find $$\int x^2\cos x \ \mathrm{d}x$$

$x^2\sin x + 2x\cos x - 2\sin x + c$

21. Find $$\int x\sin x \cos x \ \mathrm{d}x$$

$\dfrac{1}{8}(\sin 2x - 2x\cos 2x) + c$

22. Find $$\int (\tan(2x)+\cos(2x))^2 \ \mathrm{d}x$$

$\dfrac{1}{4}\left(2\tan(2x)-4\cos(2x)+\sin(2x)-2x\right)+c$

23. Find $$\int \arccos x \ \mathrm{d}x$$

$x\arccos x - \sqrt{1-x^2} + c$

24. Find $$\int \ln x \ \mathrm{d}x$$

$x\ln x - x + c$

25. Find $$\int \mathrm{e}^{2x}\sin x \ \mathrm{d}x$$

$\dfrac{\mathrm{e}^{2x}}{5}(2\sin x - \cos x) + c$

26. The curve with parametric equations $$x = 36t^2 - \pi^2 \quad y = \dfrac{1}{8}\sin 3t; \quad t \geqslant 0$$ crosses the $x$ axis at $(3\pi^2,0)$ and the $y$ axis at $(0, 0.125)$. Find the exact finite area bound by the curve and the coordinate axes.

$\pi-1$

27. Use the substitution $x = 2\sin \theta$ to find the exact value of $$\int_0^{\sqrt{2}} \dfrac{x^2}{\sqrt{4-x^2}} \ \mathrm{d}x$$

$\dfrac{\pi-2}{2}$

28. Use the substitution $x = \tan \theta$ to find the exact value of $$\int_0^1 \dfrac{1}{(1+x^2)^2} \ \mathrm{d}x$$

$\dfrac{\pi+2}{8}$

29. Use the substitution $x = \csc \theta$ to find the exact value of $$\int_{\sqrt{2}}^2 \dfrac{\sqrt{x^2-1}}{x} \ \mathrm{d}x$$

$\sqrt{3}-1-\dfrac{\pi}{12}$

30. Use integration by parts to find $$\int x^3\sqrt{9-x^2} \ \mathrm{d}x$$

$-\dfrac{1}{5}\left(9-x^2\right)^{\frac{3}{2}}(x^2+6) + c$