Integration by recognition and trigonometric identitiesEdexcel International A Level Maths: Subtopic test

1. 1
   The function $h$ is defined by $h(x)=\frac{x}{x^2+5}$ for $x\geq0$.

   (a)

   Find $\int h(x)\,dx$.

   [1 mark]

   • A$\ln(x^2+5)+c$
   • B$\frac12\ln(x^2+5)+c$
   • C$2\ln(x^2+5)+c$
   • D$-\frac{1}{x^2+5}+c$

   (b)

   Find the exact value of $\int_0^2h(x)\,dx$.

   [1 mark]

   • A$\ln\frac95$
   • B$\frac12\ln4$
   • C$\frac12\ln45$
   • D$\frac12\ln\frac95$

   (c)

   Find $\int\frac{x}{(x^2+5)^2}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=\tan x$ for $0\leq x<\frac{\pi}{2}$.

   (a)

   Find $\int f(x)\,dx$.

   [1 mark]

   • A$\ln|\cos x|+c$
   • B$\sec^2x+c$
   • C$-\ln|\cos x|+c$
   • D$\ln|\sin x|+c$

   (b)

   Find the exact value of $\int_0^{\pi/3}f(x)\,dx$.

   [1 mark]

   • A$\ln2$
   • B$-\ln2$
   • C$\frac12\ln3$
   • D$\sqrt3$

   (c)

   Find $\int[f(x)]^2\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curves $C_1$ and $C_2$ have equations $y=\cos^23x$ and $y=\sin^23x$ respectively, for $0\leq x\leq\frac{\pi}{12}$.

   (a)

   Show that $\cos^23x=\frac12(1+\cos6x)$, and hence find $\int\cos^23x\,dx$.

   [3 marks]

   (b)

   Find the exact area of the region bounded by $C_1$, $C_2$ and the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curves $C_1$ and $C_2$ have equations $y=\frac{\sin2x}{1+\cos^2x}$ and $y=\tan^2x$ respectively, for $0\leq x\leq\frac{\pi}{4}$.

   (a)

   (i) Show that $\frac{d}{dx}(1+\cos^2x)=-\sin2x$.
   (ii) Hence find the exact area of the region bounded by $C_1$, the $x$-axis and the line $x=\frac{\pi}{4}$.

   [6 marks]

   (b)

   (i) Find the exact area of the region bounded by $C_2$, the $x$-axis and the line $x=\frac{\pi}{4}$.
   (ii) Hence find the exact area of the region bounded by $C_2$, the line $y=1$ and the $y$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions