Pure: IntegrationEdexcel A-Level Maths: Topic test

1. 1
   A curve $C$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=3\sqrt{x}+\dfrac{2}{x^2}$ for $x>0$, and passes through the point $(4,10)$.

   (a)

   Which is $y$ in terms of $x$, where $c$ is a constant of integration?

   [1 mark]

   • A$\dfrac{3}{2\sqrt x}-\dfrac{4}{x^3}+c$
   • B$2x^{3/2}+\dfrac2x+c$
   • C$\dfrac32x^{3/2}-\dfrac2x+c$
   • D$2x^{3/2}-\dfrac2x+c$

   (b)

   What is the value of the constant of integration $c$?

   [1 mark]

   • A$-\frac{11}{2}$
   • B$-\frac{13}{2}$
   • C$-6$
   • D$10$

   (c)

   Find the $y$-coordinate of the point on $C$ where $x=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=6\sin2x$, where $x$ is in radians.

   (a)

   What is $\displaystyle\int f(x)\,\mathrm{d}x$?

   [1 mark]

   • A$-12\cos2x+c$
   • B$-3\cos2x+c$
   • C$3\cos2x+c$
   • D$12\cos2x+c$

   (b)

   What is the value of $\displaystyle\int_{\pi/6}^{\pi/2}f(x)\,\mathrm{d}x$?

   [1 mark]

   • A$\frac32$
   • B$6$
   • C$\frac92$
   • D$-\frac92$

   (c)

   Evaluate $\displaystyle\int_0^{\pi}f(x)\,\mathrm{d}x$ and explain why this value is not the area enclosed between the curve $y=f(x)$ and the $x$-axis for $0\le x\le\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\displaystyle\int_0^4x\sqrt{2x+1}\,\mathrm{d}x$.

   (a)

   Use the substitution $u=2x+1$ to show that $I=\dfrac14\displaystyle\int_1^9\left(u^{3/2}-u^{1/2}\right)\mathrm{d}u$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   All angles in this question are measured in radians.

   (a)

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

   [6 marks]

   (b)

   A curve satisfies the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}=x^2y\cos x$, where $y>0$, and $y=1$ when $x=0$. Use your answer to part (a) to find $y$ in terms of $x$, and find the value of $y$ when $x=\frac{\pi}{2}$, to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $f(x)=\dfrac{6x}{3x^2+1}$ for $x\ge0$.

   (a)

   What is $\displaystyle\int f(x)\,\mathrm{d}x$?

   [1 mark]

   • A$\ln(3x^2+1)+c$
   • B$\ln|6x|+c$
   • C$3\ln(3x^2+1)+c$
   • D$\dfrac{3x^2}{x^3+x}+c$

   (b)

   What is the exact value of $\displaystyle\int_0^1f(x)\,\mathrm{d}x$?

   [1 mark]

   • A$\ln2$
   • B$\ln3$
   • C$4$
   • D$\ln4$

   (c)

   Show that $\displaystyle\int_1^2f(x)\,\mathrm{d}x=\ln\dfrac{13}{4}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve satisfies the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{3y}{x}$, where $x>0$ and $y>0$.

   (a)

   Which is obtained by separating the variables?

   [1 mark]

   • A$\displaystyle\int y\,\mathrm{d}y=\int\frac3x\,\mathrm{d}x$
   • B$\displaystyle\int\frac1y\,\mathrm{d}y=\int\frac3x\,\mathrm{d}x$
   • C$\displaystyle\int\frac1y\,\mathrm{d}y=\int3x\,\mathrm{d}x$
   • D$\displaystyle\int y\,\mathrm{d}y=\int3x\,\mathrm{d}x$

   (b)

   Which is the general solution, where $A$ is a positive constant?

   [1 mark]

   • A$y=x^3+A$
   • B$y=3\ln x+A$
   • C$y=Ax^3$
   • D$y=Ax^{1/3}$

   (c)

   Find the particular solution for which $y=16$ when $x=2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curves $C_1$ and $C_2$ have equations $y=12-x^2$ and $y=x^2-6$ respectively. They intersect at the points $A$ and $B$.

   (a)

   Find the coordinates of $A$ and $B$.

   [3 marks]

   (b)

   Find the area of the finite region enclosed between $C_1$ and $C_2$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $g(x)=\dfrac{3}{x(x+3)}$ for $x>0$.

   (a)

   Express $g(x)$ in partial fractions and hence find the exact value of $\displaystyle\int_1^3g(x)\,\mathrm{d}x$.

   [6 marks]

   (b)

   The area under the curve $y=g(x)$ from $x=1$ to $x=3$ is estimated using two rectangles of equal width, each with height equal to the value of $g$ at the left-hand end of its strip. (i) Calculate the estimate. (ii) Explain why the estimate is larger than the exact area found in part (a), and state the difference to 3 significant figures. (iii) Explain how the exact area is related to rectangles of this type.

   [6 marks]

   Total for question 8: 12 marks

End of questions