Chapter 34

Determine: (a) \(\int 3 x^{2} \mathrm{~d} x\) (b) \(\int 2 t^{3} \mathrm{~d} t\)

Determine (a) \(\int 8 \mathrm{~d} x\) (b) \(\int 2 x \mathrm{~d} x\)

Determine: \(\int\left(2+\frac{5}{7} x-6 x^{2}\right) \mathrm{d} x\)

Determine: \(\int \frac{3}{x^{2}} \mathrm{~d} x\)

Determine: \(\int 3 \sqrt{x} \mathrm{~d} x\)

Determine: \(\int \frac{5}{\sqrt{x}} \mathrm{~d} x\)

Determine: (a) \(\int\left(\frac{x^{3}-2 x}{3 x}\right) \mathrm{d} x\) (b) \(\int(1-x)^{2} \mathrm{~d} x\)

Determine: (a) \(\int 5 \cos 3 x \mathrm{~d} x\) (b) \(\int 3 \sin 2 x \mathrm{~d} x\)

Determine: (a) \(\int 5 \mathrm{e}^{3 x} \mathrm{~d} x\) (b) \(\int \frac{6}{\mathrm{e}^{2 x}} \mathrm{~d} x\)

Determine: (a) \(\int \frac{3}{5 x} \mathrm{~d} x\) (b) \(\int\left(\frac{3 x^{2}-1}{x}\right) \mathrm{d} x\)

Evaluate: (a) \(\int_{1}^{2} 3 x \mathrm{~d} x\) (b) \(\int_{-2}^{3}\left(4-x^{2}\right) \mathrm{d} x\)

Evaluate: (a) \(\int_{0}^{2} x(3+2 x) \mathrm{d} x\) (b) \(\int_{-1}^{1}\left(\frac{x^{4}-5 x^{2}+x}{x}\right) \mathrm{d} x\)

Evaluate: \(\int_{0}^{\pi / 2} 3 \sin 2 x \mathrm{~d} x\)

Evaluate: \(\int_{1}^{2} 4 \cos 3 t \mathrm{~d} t\)

Determine the area enclosed by \(y=2 x+3\), the \(x\)-axis and ordinates \(x=1\) and \(x=4\).

The velocity \(\mathrm{v}\) of a body \(t\) seconds after a certain instant is: \(\left(2 t^{2}+5\right) \mathrm{m} / \mathrm{s}\). Find by integration how far it moves in the interval from \(t=0\) to \(t=4 \mathrm{~s}\).

Sketch the graph \(y=x^{3}+2 x^{2}-5 x-6\) between \(x=-3\) and \(x=2\) and determine the area enclosed by the curve and the \(x\)-axis.

Determine the area enclosed by the curve \(y=3 x^{2}+4\), the \(x\)-axis and ordinates \(x=1\) and \(x=4\) by (a) the trapezoidal rule, (b) the mid-ordinate rule, (c) Simpson's rule, and (d) integration.

Find the area enclosed by the curve \(y=\sin 2 x\), the \(x\)-axis and the ordinates \(x=0\) and \(x=\frac{\pi}{3}\),