Integration as the reverse of differentiationAQA A-Level Maths: Subtopic test

1. 1
   The gradient of the curve $C$ is given by $\frac{dy}{dx}=6x^2-4x+5$.

   (a)

   Find $\int\frac{dy}{dx}\,dx$.

   [1 mark]

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

   (b)

   The curve $C$ passes through the point $(1,3)$. Find the constant of integration $c$ in the equation of $C$.

   [1 mark]

   • A$c=3$
   • B$c=5$
   • C$c=2$
   • D$c=-2$

   (c)

   Find the value of $y$ on $C$ when $x=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ satisfies $f'(x)=3\sqrt{x}+\frac{8}{x^2}$ for $x>0$, and $f(4)=20$.

   (a)

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

   [1 mark]

   • A$2x^{\frac32}-\frac{8}{x}+c$
   • B$2x^{\frac32}+\frac{8}{x}+c$
   • C$\frac{3}{2\sqrt{x}}-\frac{16}{x^3}$
   • D$2x^{\frac32}-\frac{8}{3x^3}+c$

   (b)

   Find the constant of integration $c$ in $f(x)=2x^{\frac32}-\frac8x+c$.

   [1 mark]

   • A$c=14$
   • B$c=6$
   • C$c=34$
   • D$c=-6$

   (c)

   Find the value of $f(1)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has gradient function $\frac{dy}{dx}=\frac{(x^2+3)^2}{x^2}$ for $x>0$.

   (a)

   Find $\int\frac{dy}{dx}\,dx$.

   [3 marks]

   (b)

   The curve $C$ passes through the point $(3,20)$. Find the equation of $C$ and hence the value of $y$ when $x=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has gradient function $\frac{dy}{dx}=2kx-3$, where $k$ is a constant. $C$ passes through the points $(0,5)$ and $(2,1)$.

   (a)

   (i) Find $y$ in terms of $x$ and $k$.
   (ii) Find the value of $k$, and hence write down the equation of $C$.

   [6 marks]

   (b)

   The curve $D$ has the same gradient function as $C$, with your value of $k$, and passes through the point $(2,-3)$. Find the equation of $D$, and show that for every value of $x$ the $y$-coordinate on $C$ is $4$ greater than the $y$-coordinate on $D$. Explain why the curves are related in this way.

   [6 marks]

   Total for question 4: 12 marks

End of questions