Using any method of your choice, integrate the following functions
a) \( \int \sin ^{2} 2 x \cos ^{3} 2 x d x \)
(4 mks)
b) \( \int \frac{d x}{|x| \sqrt{9 x^{2}-1}} \)
(4 mks)
c) \( \int_{0}^{1} x \sqrt{x^{2}+1} d x \)
\( (4 \mathrm{mks}) \)
d) \( \int \sin ^{-1} a x d x \)
\( (4 \mathrm{mks}) \)
e) \( \int \ln a x, a>0 \)
(3 mks)
QUESTION TWO (Functions of several verables) \( =24 \mathrm{mks} \)
Let \( f(x, y)=x \cos y+2 y \sin x \)
1. Find the domain of the function \( f(x, y) \)
2. Find the range of the function \( f(x, y) \)
3. Find the following partials of \( f(x, y) \)
a. \( f_{x x}(x, y) \)
b. \( f_{y y}(x, y) \)
c. \( \nabla f(x, y) \)
QUESTION THREE (FOURIER/INTEGRAL) \( =6+10+8=24 \mathrm{mks} \)
1. Evaluate \( \int_{0}^{1} \int_{0}^{2} \int_{0}^{2} x^{2} y z d z d y d x \)
2. Sketch the triangular wave
\[
\begin{aligned}
\mathrm{f}(\mathrm{t}) & =\mathrm{t} \quad 0<\mathrm{t}<\pi \\
& =-\mathrm{t} \quad-\pi<\mathrm{t}<0 \\
\mathrm{f}(\mathrm{t}) & =\mathrm{f}(\mathrm{t}+2 \pi)
\end{aligned}
\]
for \( -5 \pi<t<5 \pi \)
