91Ó°ÊÓ

Determine the following integrals: \(\mathbf{a} \int b\left(b^{2}-3\right)^{7} \mathrm{~d} b\) \(\mathbf{b} \int(5 s-1)^{9} \mathrm{~d} s\) \(c \int\left(3 a^{2}-4 a\right)\left(a^{3}-2 a^{2}+6\right)^{4} d a\) d \(\int 21 q^{2} \sqrt{7 q^{3}-5} \mathrm{~d} q\) e \(\int \frac{p^{2}-1}{\sqrt{p^{3}-3 p}} \mathrm{~d} p\) f \(\int \frac{\alpha-1}{\left(\alpha^{2}-2 \alpha+10\right)^{2}} \mathrm{~d} \alpha\)

The average power, \(P\), of an a.c. circuit is given by $$ P=\frac{\omega}{2 \pi} \int_{0}^{\frac{2 \pi}{u}}\left(i^{2} R\right) \mathrm{d} t $$ where \(i=I \sin (\omega t), \omega\) is angular frequency, \(I\) is peak current, \(R\) is resistance and \(t\) is time. Show that $$ P=\frac{I^{2} R}{2} $$

Determine the following integrals: a \(\int 2 x e^{x} \mathrm{~d} x\) \(\mathbf{b} \int t \sin (t) \mathrm{d} t\)

Determine a \(\int \frac{3 c+4}{(c+1)(c+2)} d c\) b \(\int \frac{2 \lambda}{\lambda^{2}-1} \mathrm{~d} \lambda\) c \(\int \frac{2 a+7}{a^{2}+a-2} \mathrm{~d} a\) d \(\int \frac{-12 y-13}{(2 y+1)(y-3)} \mathrm{d} y\)

Determine the following: \(\mathbf{a} \int \sin (t) \mathrm{d} t\) \(\mathbf{b} \int \cos (t) \mathrm{d} t\) \(\mathbf{c} \int \tan (t) \mathrm{d} t\) \(\mathbf{d} \int e^{t} \mathrm{~d} t\) e \(\int \cosh (t) \mathrm{d} t \quad \mathbf{f} \int 9.81 \mathrm{~d} t\) \(\mathbf{g} \int 25 \mathrm{~d} x \quad \mathbf{h} \int \frac{1}{2} \mathrm{~d} x\)

Obtain a \(\int \sin (7 x+1) \mathrm{d} x\) b \(\int \cos (7 x+1) \mathrm{d} x\)

[reliability engineering] The mean time to failure, MTTF, in years, for a set of components is given by $$ \mathrm{MTTF}=\int_{0}^{\mathrm{s}}(1-0.2 t)^{1.5} \mathrm{~d} t $$ Evaluate MTTF.

Find \(\mathbf{a} \int \cos (\omega t) \mathrm{d} t\) \(\mathbf{b} \int \cos (\omega t+\theta) \mathrm{d} t\)

A particle moves along a horizontal line with displacement, \(s\), given by $$ s=\int_{0}^{2}\left(t^{2}-2 t\right) d t $$ Determine \(s\).

Find \({ }^{*} \mathbf{a} \int \cos (\omega t) \mathrm{d}(\omega t)\) b \(\int 10^{x} \mathrm{~d} x\) \(\mathbf{c} \int \sinh (t) \mathrm{d} t\) \(\mathbf{d} \int \sec (x) \mathrm{d} x\)