Integrating standard functionsEdexcel A-Level Maths: Subtopic test

1. 1
   A curve has gradient $\frac{\mathrm{d}y}{\mathrm{d}x}=6e^{2x}-\frac{4}{x}$ for $x>0$, and passes through the point $(1,\,3e^2+1)$.

   (a)

   Find $\int 6e^{2x}\,\mathrm{d}x$.

   [1 mark]

   • A$6e^{2x}+c$
   • B$3e^{2x}+c$
   • C$12e^{2x}+c$
   • D$\frac{6e^{2x+1}}{2x+1}+c$

   (b)

   Find $\int -\frac{4}{x}\,\mathrm{d}x$ for $x>0$.

   [1 mark]

   • A$-4\ln x+c$
   • B$\frac{2}{x^2}+c$
   • C$4\ln x+c$
   • D$-\frac{\ln x}{4}+c$

   (c)

   Find the equation of the curve.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Angles are in radians. A function is defined by $f(x)=\sin3x+\cos\frac{x}{2}$.

   (a)

   Find $\int\sin3x\,\mathrm{d}x$.

   [1 mark]

   • A$\frac13\cos3x+c$
   • B$-3\cos3x+c$
   • C$-\frac13\cos3x+c$
   • D$3\cos3x+c$

   (b)

   Find $\int\cos\frac{x}{2}\,\mathrm{d}x$.

   [1 mark]

   • A$\frac12\sin\frac{x}{2}+c$
   • B$-2\sin\frac{x}{2}+c$
   • C$\sin\frac{x}{2}+c$
   • D$2\sin\frac{x}{2}+c$

   (c)

   Find the exact value of $\int_0^{\pi}f(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are in radians. Trigonometric identities are used to rewrite an expression before it is integrated.

   (a)

   Show that $\int\sin^2x\,\mathrm{d}x=\frac{x}{2}-\frac{\sin2x}{4}+c$.

   [3 marks]

   (b)

   Find the exact value of $\int_0^{\pi/4}\tan^2x\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Angles are in radians throughout this question.

   (a)

   (i) Show that $\cos^23x=\frac12(1+\cos6x)$.
   (ii) Hence find the exact value of $\int_0^{\pi/6}\cos^23x\,\mathrm{d}x$.

   [6 marks]

   (b)

   Show that $\int_0^{\pi/8}\left(4\sec^22x+6\sin2x\right)\mathrm{d}x=\frac{10-3\sqrt2}{2}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions