1.08h Integration by substitution

14 The curve \(C\) is defined for \(t \geq 0\) by the parametric equations $$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$ \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717} 14

1. Find the gradient of \(C\) at the point where it intersects the positive \(x\)-axis.
   14
2. (i) The area \(A\) enclosed between \(C\) and the \(x\)-axis is given by $$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$ Find the value of \(b\).
   14 (b) (ii) Use the substitution \(y = 4 t ^ { 2 } - t ^ { 3 }\) to show that $$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$ 14 (b) (iii) Find the value of \(A\).

15

1. Given that $$y = \operatorname { cosec } \theta$$ 15
   1. (i) Express \(y\) in terms of \(\sin \theta\). 15
   2. (ii) Hence, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$ 15
   3. (iii) Show that $$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$ 15
   4. 1. Use the substitution $$x = 2 \operatorname { cosec } u$$ to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x \quad \text { for } x > 2$$ can be written as $$k \int \sin u \mathrm {~d} u$$ where \(k\) is a constant to be found.
         15
   5. (ii) Hence, show $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 4 } } { 4 x } + c \quad \text { for } x > 2$$ where \(c\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-32_2492_1721_217_150}
      \includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-36_2496_1721_214_148}

11 By using the substitution \(u = 1 + \sqrt { x }\), find \(\int \frac { x } { 1 + \sqrt { x } } \mathrm {~d} x\). Answer all the questions.

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\) The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\) [0pt] [5 marks]
      12
      1. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
         [0pt] [4 marks] 12
   2. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
      [0pt] [2 marks]
      12
   3. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
      [0pt] [1 mark]

5
Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.

1. (a) Show that the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) transforms the integral

$$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$ into the integral $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$ (b) Hence determine $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$

1. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that

$$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + c$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.

1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.

4 Find

1. \(\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x\)
2. \(\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\)

7

1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).

10

1. Show that \(\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)\).
2. Find \(\int x \sqrt { x - 2 } \mathrm {~d} x\).

11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).

4

1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).

10

1. By using the substitution \(u = 3 - 2 x\), or otherwise, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3\).
2. \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-4_595_588_927_817} The diagram shows the region \(R\), which is bounded by the curve \(y = \frac { 4 x } { 3 - 2 x }\), the \(y\)-axis and the line \(y = 4\). Find the exact volume generated when the region \(R\) is rotated completely around the \(x\)-axis. {www.cie.org.uk} after the live examination series. }

4

1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).

6

1. (a) Using the substitution \(u = \frac { 1 } { 2 } \pi - x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } u \mathrm {~d} u$$ (b) Hence find the common value of these definite integrals.
2. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x$$

a) Find \(\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x\). b) Find \(\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x\).
c) Find \(\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x\).
d) Use the substitution \(u = 2 \cos x + 1\) to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$

a) Use a suitable substitution to find $$\int \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$$ b) Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x\). END OF PAPER \end{document}

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = \frac{10}{2x + 5\sqrt{x}}\), \(x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the lines with equations \(x = 1\) and \(x = 4\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac{10}{2x + 5\sqrt{x}}\)

1. [(a)] Complete the table above by giving the missing value of \(y\) to 5 decimal places. \hfill [1]
2. [(b)] Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places. \hfill [3]
3. [(c)] By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of \(R\). \hfill [1]
4. [(d)] Use the substitution \(u = \sqrt{x}\), or otherwise, to find the exact value of $$\int_1^4 \frac{10}{2x + 5\sqrt{x}} dx$$ \hfill [6]

\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\). [5]
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]

Use the substitution \(2x = \tan \theta\) to find the exact value of $$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$ Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are rational numbers. [9]

\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
2. Find the exact \(x\)-coordinate of \(M\). [6]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]