Further integration: practice Problem set B

\(\frac{dy}{dx}=21\sqrt{x}\), and y = 400 when x = 9. Find y in terms of x.

Find the required function in each case.

1. (a) \(f'(x)=4\cos x-7\sin x\) and f(0) = 10. Find f(x).
2. (b) \(\frac{dy}{dx}=5e^x-\frac{2}{x}\) for x > 0, and y = 1 when x = 1. Find y in terms of x.
3. (c) \(\int\frac{8x^2+6}{2x}\,dx\)

Find each integral.

1. (a) \(\int(2-5x)^{7}\,dx\)
2. (b) \(\int(5\sin(3x)+4\cos(6x))\,dx\)
3. (c) \(\int 4x(x^2+2)^{6}\,dx\)

Find each integral.

1. (a) \(\int x^2\sqrt{3x^3+7}\,dx\)
2. (b) \(\int x\sqrt[3]{5-3x^2}\,dx\)

Find each integral.

1. (a) \(\int 14xe^{4x^2}\,dx\)
2. (b) \(\int 9x^4e^{-3x^5}\,dx\)

Find each integral.

1. (a) \(\int\frac{x^4}{3x^5+2}\,dx\)
2. (b) \(\int\frac{x}{5-2x^2}\,dx\)

Use a substitution to find the following.

1. (a) \(\frac{dy}{dx}=x\cos(4x^2)\). Find y in terms of x.
2. (b) \(\int\frac{6x}{\sqrt{3x^2+1}}\,dx\)

Air is pumped into a balloon. After t seconds the volume of air is V cm³, and \(\frac{dV}{dt}=0.9t^2+3\) and V = 20 when t = 0.

1. (a) Find an expression for \(V\) in terms of t.
2. (b) Find the volume when t = 4.

The depth of a submarine below the surface is x metres at time t minutes, and \(\frac{dx}{dt}=15-4e^t\) and x = 30 when t = 0.

1. (a) Find an expression for \(x\) in terms of t.
2. (b) Find the depth when t = 2.

Find f(x) in each case.

1. (a) \(f'(x)=\frac{4}{(3x-2)^2}\), and the graph of y = f(x) passes through the point (2, 1). Find f(x).
2. (b) \(f'(x)=\frac{3\ln x}{4x}\) for x > 0, and the curve y = f(x) passes through the point \((e^2, 6)\). Find f(x).

\(f'(x)=\frac{5}{x}\) and f(-3) = 2. Find the exact value of \(f(-7)\).

Find each integral.

1. (a) \(\int\left(10\cos^4 x\sin x+kx^{2}+3\right)dx\), where k is a constant.
2. (b) \(\int\frac{e^{3x}+e^{-x}}{e^{x}}\,dx\)

Find each integral.

1. (a) \(\int\sqrt{x^4+8x^2}\,dx\) for x > 0.
2. (b) \(\int\frac{4}{x\ln x}\,dx\) for x > 1.

Find each integral.

1. (a) Given that \(\cos 2x=1-2\sin^2x\), find \(\int 8\sin^2x\,dx\).
2. (b) \(\int\tan 4x\,dx\), showing your working.
3. (c) \(\int 3\cos^3x\,dx\)