IBDP Past Year Exam Questions – Techniques for Integration

1. [M08/P2/TZ1]

By using an appropriate substitution find 

               $\int \frac{\tan \left(\ln y\right)}{y}dy, y>0$

2. [M98/P1]

Find     $\int arc\tan x dx$ .

3. [N95/P1]

Find  $\int x^2{e}^{–x} dx$ .

4. [M11/P2/TZ2]

Using integration by parts, show that  $\int e^{2x}\sin x dx=\frac{1}{5}{e}^{2x}\left(2\sin x –\cos x\right)+C$ . [6 marks]

5. [N12/P2/TZ2]

By using the substitution  $x=\sin t$ , find  $\int \frac{x^3}{\sqrt{1–x^2}}dx$ .    [7 marks]

6. [M13/P1/TZ1]

The function  $f$ is defined by  $f\left(x\right)=\frac{1}{4x^2–4x+5}$ . By using a suitable substitution show that  $\int f\left(x\right) dx=\frac{1}{4}\int \frac{1}{u^2+1}du$ .

7. [N13/P2/TZ0]

By using the substitution  $x=2 \tan u$ , show that 

                            $\int \frac{dx}{x^2\sqrt{x^2+4}}=\frac{–\sqrt{x^2+4}}{4x}+C$         [7 marks]

8. [N14/P1/TZ0]

By using the substitution  $u=1+\sqrt{x}$ , find  $\int \frac{\sqrt{x}}{1+\sqrt{x}} dx$ .   [6 marks]

9. [M16/P2/TZ2]

10. [N17/P2/TZ0]

By using the substitution  $x^2=2 sec \theta$ , show that  $\int \frac{dx}{x\sqrt{x^4–4}}=\frac{1}{4}arc\cos \left(\frac{2}{x^2}\right)+c$ .     [7 marks]