Further integration: practice Problem set A

\(\frac{dy}{dx}=15\sqrt{x}\), and y = 75 when x = 4. Find y in terms of x.

Find the required function in each case.

1. (a) \(f'(x)=6\cos x-5\sin x\) and f(0) = 9. Find f(x).
2. (b) \(\frac{dy}{dx}=3e^x+\frac{4}{x}\) for x > 0, and y = 2 when x = 1. Find y in terms of x.
3. (c) \(\int\frac{9x^2+4}{3x}\,dx\)

Find each integral.

1. (a) \(\int(7-3x)^{8}\,dx\)
2. (b) \(\int(3\sin(4x)+8\cos(2x))\,dx\)
3. (c) \(\int 6x(x^2+3)^{4}\,dx\)

Find each integral.

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

Find each integral.

1. (a) \(\int 9xe^{6x^2}\,dx\)
2. (b) \(\int 8x^2e^{-2x^3}\,dx\)

Find each integral.

1. (a) \(\int\frac{x^2}{2x^3+9}\,dx\)
2. (b) \(\int\frac{x^3}{1-x^4}\,dx\)

Use a substitution to find the following.

1. (a) \(\frac{dy}{dx}=x\sin(5x^2)\). Find y in terms of x.
2. (b) \(\int\frac{4x}{\sqrt{2x^2+9}}\,dx\)

Water flows into a tank. After t minutes the volume of water is V litres, and \(\frac{dV}{dt}=0.6t^2+2\) and V = 15 when t = 0.

1. (a) Find an expression for \(V\) in terms of t.
2. (b) Find the volume when t = 5.

The height of a drone above the ground is x metres at time t seconds, and \(\frac{dx}{dt}=12-5e^t\) and x = 8 when t = 0.

1. (a) Find an expression for \(x\) in terms of t.
2. (b) Find the height when t = 1.5.

Find f(x) in each case.

1. (a) \(f'(x)=\frac{3}{(2x+1)^2}\), and the graph of y = f(x) passes through the point (1, 0). Find f(x).
2. (b) \(f'(x)=\frac{5\ln x}{2x}\) for x > 0, and the curve y = f(x) passes through the point \((e, 3)\). Find f(x).

\(f'(x)=\frac{3}{x}\) and f(-2) = 4. Find the exact value of \(f(-5)\).

Find each integral.

1. (a) \(\int\left(8\sin^3 x\cos x+kx^{3}+2\right)dx\), where k is a constant.
2. (b) \(\int\frac{e^{5x}-e^{-x}}{e^{2x}}\,dx\)

Find each integral.

1. (a) \(\int\sqrt{x^4+5x^2}\,dx\) for x > 0.
2. (b) \(\int\frac{6}{x\ln x}\,dx\) for x > 1.

Find each integral.

1. (a) Given that \(\cos 2x=2\cos^2x-1\), find \(\int 6\cos^2x\,dx\).
2. (b) \(\int\tan 3x\,dx\), showing your working.
3. (c) \(\int 6\sin^3x\,dx\)