Indefinite integrationEdexcel International A Level Maths: Subtopic test

1. 1
   Let $g(x)=6x^2-4x+5$.

   (a)

   Find $\int g(x)\,\mathrm{d}x$.

   [1 mark]

   • A$12x-4+c$
   • B$2x^3-2x^2+5+c$
   • C$6x^3-4x^2+5x+c$
   • D$2x^3-2x^2+5x+c$

   (b)

   Why must a constant of integration $+c$ be included in $\int g(x)\,\mathrm{d}x$?

   [1 mark]

   • ASo that the answer is always positive
   • BA constant differentiates to zero, so many different functions have the derivative $g(x)$
   • CBecause $g(x)$ contains the constant term $5$
   • DSo that the integral equals zero when $x=0$

   (c)

   Find $\int\left(g(x)+\frac{4}{x^3}\right)\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function is defined by $f(x)=4\sqrt{x}-\dfrac{6}{x^3}$ for $x>0$.

   (a)

   Which of the following is $f(x)$ written as a sum of powers of $x$?

   [1 mark]

   • A$4x^{\frac12}-6x^{3}$
   • B$4x^{2}-6x^{-3}$
   • C$4x^{\frac12}-6x^{-3}$
   • D$4x^{-\frac12}-6x^{-3}$

   (b)

   Find $\int f(x)\,\mathrm{d}x$.

   [1 mark]

   • A$\frac83x^{\frac32}+3x^{-2}+c$
   • B$\frac83x^{\frac32}-3x^{-2}+c$
   • C$6x^{\frac32}+3x^{-2}+c$
   • D$\frac83x^{\frac32}+\frac32x^{-4}+c$

   (c)

   Find $\int x^3f(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=\dfrac{(2x+1)^2}{\sqrt{x}}$ for $x>0$.

   (a)

   Show that $f(x)=4x^{\frac32}+4x^{\frac12}+x^{-\frac12}$.

   [3 marks]

   (b)

   Hence find $\int f(x)\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student is practising integration for $x>0$ and writes $\displaystyle\int\left(x+\dfrac{4}{x^{2}}\right)\mathrm{d}x=\dfrac{x^{2}}{2}+\dfrac{4}{x}+c$.

   (a)

   Identify the error in the student's working, give the correct integral and verify it by differentiating.

   [6 marks]

   (b)

   The student's next exercise is to find $\displaystyle\int\left(\dfrac{x^{3}-2x}{\sqrt{x}}+\dfrac{9}{x^{4}}\right)\mathrm{d}x$. Find this integral.

   [6 marks]

   Total for question 4: 12 marks

End of questions