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

1. 1
   A curve has gradient function $\frac{dy}{dx}=6x^{2}-4x+5$ and passes through the point $(1,7)$.

   (a)

   Find $\int\left(6x^{2}-4x+5\right)dx$.

   [1 mark]

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

   (b)

   Find the value of the constant of integration for the curve.

   [1 mark]

   • A$7$
   • B$-2$
   • C$12$
   • D$2$

   (c)

   Find the $y$-coordinate of the point on the curve where $x=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $y=f(x)$, for $x>0$, has gradient function $f'(x)=\frac{(x-1)^{2}}{\sqrt x}$ and passes through the point $(1,2)$.

   (a)

   Which expression is equal to $\frac{(x-1)^{2}}{\sqrt x}$?

   [1 mark]

   • A$x^{3/2}-2x^{1/2}+x^{-1/2}$
   • B$x^{3/2}-x^{-1/2}$
   • C$x^{5/2}-2x^{3/2}+x^{1/2}$
   • D$x^{3/2}-2x^{1/2}-x^{-1/2}$

   (b)

   Find $\int\frac{(x-1)^{2}}{\sqrt x}\,dx$.

   [1 mark]

   • A$\frac32x^{1/2}-x^{-1/2}-\frac12x^{-3/2}+c$
   • B$\frac25x^{5/2}-\frac43x^{3/2}+2x^{1/2}+c$
   • C$\frac25x^{5/2}-\frac43x^{3/2}+\frac12x^{1/2}+c$
   • D$\frac25x^{5/2}-\frac43x^{3/2}+2x^{-1/2}+c$

   (c)

   Find an equation of the curve.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $y=f(x)$, for $x>0$, has gradient function $f'(x)=3x^{2}-\frac{6}{x^{3}}$ and passes through the point $(1,2)$.

   (a)

   Find $f(x)$.

   [3 marks]

   (b)

   Find an equation of the tangent to the curve at the point where $x=2$, in the form $y=mx+c$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$, defined for $x<0$, has gradient function $\frac{dy}{dx}=\frac{(x+2)^{2}}{x^{4}}$ and passes through the point $(-1,1)$.

   (a)

   Find $y$ in terms of $x$.

   [6 marks]

   (b)

   Find the coordinates of the stationary point of $C$, and deduce that it is neither a maximum nor a minimum.

   [6 marks]

   Total for question 4: 12 marks

End of questions