1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

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1. Using the quotient rule, show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tan 3 \theta ) = 3 + 3 \tan ^ { 2 } 3 \theta\) for \(- \frac { 1 } { 6 } \pi < \theta < \frac { 1 } { 6 } \pi\).
2. Hence find the value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 9 } \pi } \tan ^ { 2 } 3 \theta \mathrm {~d} \theta\), giving your answer in the simplest exact form.

Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$ a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\). The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
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0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\). Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

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 $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$

The equation of a curve is such that \(\frac{dy}{dx} = 4x - 3\sqrt{x} + 1\).

1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac{11}{2}\). [3]
2. Given that the curve passes through the point \((4, 11)\), find the equation of the curve. [4]

\includegraphics{figure_7} The diagram shows part of the curve with equation \(y = \frac{12}{\sqrt{2x+1}}\). The point \(A\) on the curve has coordinates \(\left(\frac{7}{2}, 6\right)\).

1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = mx + c\). [4]
2. Find the area of the region bounded by the curve and the lines \(x = 0\), \(x = \frac{7}{2}\) and \(y = 0\). [4]

A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).

1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
2. Find f\((x)\). [4]

It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$

1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]

\includegraphics{figure_2} The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]

Showing all necessary working, find \(\int_1^4 \left(\sqrt{x} + \frac{2}{\sqrt{x}}\right) \text{d}x\). [4]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
2. Find the coordinates of \(M\). [3]
3. Find, showing all necessary working, the area of the shaded region. [4]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at A. [3]
2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]

Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]

It is given that \(\int_1^a \left(\frac{4}{1 + 2x} + \frac{3}{x}\right) dx = \ln 10\), where \(a\) is a constant greater than 1.

1. Show that \(a = \sqrt{90(1 + 2a)^{-2}}\). [5]
2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures. [3]

It is given that \(\int_a^{a^2} \frac{10}{2x+1} dx = 7\), where \(a\) is a constant greater than \(1\).

1. Show that \(a = \sqrt[9]{0.5e^{1.4}(2a+1) - 0.5}\). [5]
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to \(3\) significant figures. Use an initial value of \(2\) and give the result of each iteration to \(5\) significant figures. [3]

\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\). [3]
2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]

1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]

By using appropriate trigonometrical identities, find the exact value of

1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]

It is given that the positive constant \(a\) is such that $$\int_{-a}^a (4e^{2x} + 5) dx = 100.$$

1. Show that \(a = \frac{1}{4}\ln(50 + e^{-2a} - 5a)\). [4]
2. Use the iterative formula \(a_{n+1} = \frac{1}{4}\ln(50 + e^{-2a_n} - 5a_n)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]

Show that \(\int_1^7 \frac{6}{2x + 1} \, dx = \ln 125\). [5]

\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\). [4]
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]