Inverse trigonometric functions: differentiation and integrationEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $f(x)=\frac12\arctan\left(x^2\right)$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\frac{1}{2\left(1+x^4\right)}$
   • B$\frac{x}{1+x^4}$
   • C$\frac{x}{\left(1+x^2\right)^2}$
   • D$\frac{1}{1+x^4}$

   (b)

   Find the value of $f'(1)$.

   [1 mark]

   • A$1$
   • B$\frac14$
   • C$\frac\pi8$
   • D$\frac12$

   (c)

   Hence find the exact value of $\int_0^1\frac{x}{1+x^4}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $y=\arcsin x+x\sqrt{1-x^2}$ for $-1<x<1$.

   (a)

   Find the derivative of $x\sqrt{1-x^2}$ with respect to $x$.

   [1 mark]

   • A$\sqrt{1-x^2}-\frac{x^2}{\sqrt{1-x^2}}$
   • B$\sqrt{1-x^2}+\frac{x^2}{\sqrt{1-x^2}}$
   • C$\frac{1}{\sqrt{1-x^2}}$
   • D$-\frac{x}{\sqrt{1-x^2}}$

   (b)

   Find $\frac{dy}{dx}$ in its simplest form.

   [1 mark]

   • A$\frac{1}{\sqrt{1-x^2}}+\sqrt{1-x^2}$
   • B$\frac{2}{\sqrt{1-x^2}}$
   • C$2\sqrt{1-x^2}$
   • D$\sqrt{1-x^2}$

   (c)

   Hence find the exact value of $\int_0^{1/2}\sqrt{1-x^2}\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{1}{\sqrt{9-4x^2}}$ and $g(x)=\frac{1}{9+4x^2}$.

   (a)

   Use the substitution $x=\frac32\sin\theta$ to show that $\int_0^{3/4}f(x)\,dx=\frac\pi{12}$.

   [3 marks]

   (b)

   Find the exact value of $\int_0^{3/2}g(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $I=\int_0^1\sqrt{4-x^2}\,dx$ and $J=\int_0^1\frac{1}{\left(1+x^2\right)^2}\,dx$.

   (a)

   Use the substitution $x=2\sin\theta$ to find the exact value of $I$.

   [6 marks]

   (b)

   Use the substitution $x=\tan\theta$ to show that $J=\frac\pi8+\frac14$.

   [6 marks]

   Total for question 4: 12 marks

End of questions