# IBDP Past Year Exam Questions – Definite Integrals

###### 1.   [M18/P1/TZ1]

Given that  and  , find

(a)  ${\int }_{–2}^{0}f\left(\left(x\right)+2\right)dx$ ;     (b)   ${\int }_{–2}^{0}f\left(x+2\right)dx$ .    [6 marks]

###### 2.   [M18/P1/TZ2]

(a)  Use the substitution  $u={x}^{1}{2}}$ to find  $\int \frac{dx}{{x}^{3}{2}}+{x}^{1}{2}}}$ .

(b)  Hence find the value of  $\frac{1}{2}{\int }_{1}^{9}\frac{dx}{{x}^{3}{2}}+{x}^{1}{2}}}$ , expressing your answer in the form of arctan $q$ , where  $q\in \mathrm{ℚ}$ .                                   [7 marks]

###### 3.   [M17/P1/TZ2]

(a)   Using the substitution $x=\mathrm{tan}\theta$ show that  ${\int }_{0}^{1}\frac{1}{{\left({x}^{2}+1\right)}^{2}}dx={\int }_{0}^{\mathrm{\pi }}{4}}{\mathrm{cos}}^{2}\theta d\theta$

(b)   Hence find the value of   ${\int }_{0}^{1}\frac{1}{{\left({x}^{2}+1\right)}^{2}}dx$ .       [7 marks]

###### 4.   [M16/P1/TZ2]

(a)  Show that  for  $0<\alpha <\frac{\mathrm{\pi }}{2}$ .

(b)   Hence find  .              [5 marks]

###### 5.   [M15/P1/TZ2]

Show that  ${\int }_{1}^{2}{x}^{3}\mathrm{ln}xdx=4\mathrm{ln}2–\frac{15}{16}$ .

###### 6.   [N99/P1]

Find the real number  $k>1$ for which  ${\int }_{1}^{k}\left(1+\frac{1}{{x}^{2}}\right)dx=\frac{3}{2}$ .