%M148 W2015 A2
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\begin{center}
\textbf{\large Math 148 Assignment 2}\\[1ex]
\textbf{Due  2:00 p.m.\ Friday, January 30 in the Math 148 dropbox. }  
\end{center}

\vskip 2em 
\begin{Ex}

\item   Compute the following integrals: \\[2ex]
(a) $\dint \dfrac{\sin^3 x}{\sqrt{\cos x}} \,dx$ \qquad\quad
(b) $\dint x^2 \sin^{-1}(x^3) \,dx $ \qquad\quad
(c)$\dint_0^{63} \dfrac{dt}{\sqrt{1+t}+\sqrt[3]{1+t}}$

\item  Compute the following integrals: \\[2ex]
(a) $\dint_1^2 \bigl(\log x \bigr)^2 \,dx$ \qquad\quad
(b) $\dint e^{2x} \cos(3x) \,dx$.\qquad\quad
(c) $\dint_{\!-1}^1 x^3 e^{x^4} \cos 2x \,dx$

\item  Compute the following integrals: \\[2ex]
(a) $\dint \dfrac{5x^2-13x+9}{x^3-3x^2+4} \,dx$ \quad
(b) $\dint_{\!-3}^{-2} \dfrac{x^2+8x+10}{(x^2+6x+10)^2} \,dx$ \quad
(c) $\dint_{\!-\pi/2}^{\pi/2} \dfrac{1}{5+\sin x + 7\cos x} \,dx .$

\item \begin{parts}
\item Compute a recursion formula for 
 $I_m = \dint x^a (\log x)^m \,dx$, $m\ge0$ and $a \ne -1$.\\[1ex]
 Hence obtain an explicit formula for $I_3$.

\item 
$\dint_0^\pi \dfrac{x \sin x}{1 + \cos^2 x} \,dx$ \qquad  
\hint Substitute $u = \pi - x$ and combine the two integrals.
\end{parts} 

\item Suppose that $f(x)$ is a $C^2$ function on $\bR$ such that $|f(x)| \le A$ and $|f''(x)| \le C$ for $x \in \bR$.
Prove that $|f'(x)| \le \sqrt{2AC}$.\\[1.5ex]
\hint fix $x_0$ with $f'(x_0) = b \ge 0$. Get a lower bound for $f'(x_0 \pm h)$.\\[.5ex]
Use this to estimate $\dint_{x_0-H}^{x_0+H} f'(x) \,dx$ for a good choice of $H$.
 
 
\item Suppose that $f(0)=0$ and $0<f'(x)\le 1$ for all $x\ge0$.
Show that
\[
  \int_0^x f(t)^3 \,dt \le \bigg( \int_0^x f(t) \,dt \bigg)^2 
 \qquad \text{for all  } x>0 .
\]
When does equality hold? \\
\hint differentiate, factor and differentiate again.

\end{Ex}
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