Integration by substitution Problem set A

Use the given substitution to find each integral.

1. (a) \(\int \sin 9x\,dx,\quad u=9x\)
2. (b) \(\int \frac{1}{7-3x}\,dx,\quad u=7-3x\)
3. (c) \(\int e^{2+5x}\,dx,\quad u=2+5x\)

Use the given substitution to find each integral.

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

Use the given substitution to find each integral.

1. (a) \(\int x\sqrt{x^2+7}\,dx,\quad u=x^2+7\)
2. (b) \(\int \sin x\sqrt{\cos x}\,dx,\quad u=\cos x\)

Use the given substitution to find each integral.

1. (a) \(\int \frac{\cos\sqrt x}{\sqrt x}\,dx,\quad u=\sqrt x\)
2. (b) \(\int \frac{\sin(\ln x)}{x}\,dx,\quad u=\ln x\)

Choose a substitution to find each integral.

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

Choose a substitution to find each integral.

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

Choose a substitution to find each integral. In the second part, write tan as sin over cos.

1. (a) \(\int \sin^3x\cos x\,dx\)
2. (b) \(\int \tan 2x\,dx\)

A particle moves in a straight line with velocity \(v=t\sqrt{t^2+9}\) m s\(^{-1}\), where t is the time in seconds. Find the distance travelled in the first 4 seconds.

A hot drink cools. Its temperature T, in °C, at time t minutes satisfies \(\frac{dT}{dt}=-12e^{-0.3t}\). At \(t=0\), \(T=90\). Find T when \(t=5\).

The cross-section of a ramp is the region under \(y=\frac{x}{4+x^2}\) from \(x=0\) to \(x=2\) (units in metres). Find its area, then check with your calculator.

Use \(u=x+3\) to find \(\int\sqrt{x^3+3x^2}\,dx\), for \(x>0\).

Use \(x=\sin u\) to evaluate \(\int_0^{\sqrt3/2}\frac{1}{\sqrt{1-x^2}}\,dx\).