Integration by substitution Problem set B

Use the given substitution to find each integral.

1. (a) \(\int \cos 7x\,dx,\quad u=7x\)
2. (b) \(\int \frac{1}{4-5x}\,dx,\quad u=4-5x\)
3. (c) \(\int e^{3+2x}\,dx,\quad u=3+2x\)

Use the given substitution to find each integral.

1. (a) \(\int \frac{e^x}{6+e^x}\,dx,\quad u=6+e^x\)
2. (b) \(\int \frac{x^3}{2+x^4}\,dx,\quad u=2+x^4\)

Use the given substitution to find each integral.

1. (a) \(\int x\sqrt{x^2+11}\,dx,\quad u=x^2+11\)
2. (b) \(\int \cos x\sqrt{1+\sin x}\,dx,\quad u=1+\sin x\)

Use the given substitution to find each integral.

1. (a) \(\int \frac{e^{-\sqrt x}}{\sqrt x}\,dx,\quad u=\sqrt x\)
2. (b) \(\int \frac{1}{x(1+\ln x)}\,dx,\quad u=1+\ln x\)

Choose a substitution to find each integral.

1. (a) \(\int \sin 8x\,dx\)
2. (b) \(\int e^{-4x}\,dx\)
3. (c) \(\int \frac{1}{2+5x}\,dx\)
4. (d) \(\int \frac{1}{9-x}\,dx\)

Choose a substitution to find each integral.

1. (a) \(\int x^7e^{x^8}\,dx\)
2. (b) \(\int x^3\cos(x^4)\,dx\)
3. (c) \(\int \frac{x}{5+4x^2}\,dx\)
4. (d) \(\int \frac{\sin x}{3+\cos x}\,dx\)

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

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

Oil flows into a tank at a rate of \(R=t\sqrt{t^2+16}\) litres per minute, where t is the time in minutes. Find the volume of oil that flows in during the first 3 minutes.

The concentration C of a drug in the blood, in mg per litre, at time t hours satisfies \(\frac{dC}{dt}=-8e^{-0.2t}\). At \(t=0\), \(C=30\). Find C when \(t=6\).

A garden bed is the region under \(y=\frac{e^x}{5+e^x}\) from \(x=0\) to \(x=2\) (units in metres). Find its area, then check with your calculator.

Use \(u=x+5\) to find \(\int\sqrt{x^3+5x^2}\,dx\), for \(x>0\).

Use \(x=\sin u\) to evaluate \(\int_0^{\sqrt2/2}\frac{1}{\sqrt{1-x^2}}\,dx\).