\ANG{
\begin{tabular}[t]{lll}
(a) $\D\int\frac{x^3}{5x^4+3}\,\RMd x$ & 
(b) $\D\int \frac{x^{n-1}}{x^n+a}\,\RMd x$ & 
(c) $\D\intop_2^3 \frac{x^2}{x^3-1}\,\RMd x$ 
\rule[-5mm]{0mm}{12mm}\\
(d) $\D\intop_\e^{\e^2}\frac{\RMd x}{x\ln x}$ &
(e) $\D\intop_{0}^{\frac{\pi}2} \frac{\sin x-\cos x}{\sin x+\cos x}\,\RMd x$ & 
(f)  $\D\intop_{0}^{\frac{\pi}4} \frac{\sin x-\cos x}{\sin x+\cos x}\,\RMd x$
\end{tabular}
} 
\LSG{
\begin{enumerate}
\item $\D\int\frac{x^3}{5x^4+3}\,\RMd x=
\frac 1{20}\int\frac{20x^3}{5x^4+3}\,\RMd x=
\frac 1{20}\int\frac{(5x^4+3)'}{5x^4+3}\,\RMd x=
\frac 1{20}\ln|5x^4+3|+C$ 
%%
\item $\D\int\frac{x^{n-1}}{x^n+a}\,\RMd x=
\frac 1{n}\int\frac{nx^{n-1}}{x^n+a}\,\RMd x=
\frac 1{n}\int\frac{(x^n+a)'}{x^n+a}\,\RMd x=
\frac 1{n}\ln|x^n+a|+C$ 
%%
\item $\D\intop_2^3\frac{x^2}{x^3-1}\,\RMd x=
\frac 13\intop_2^3\frac{3x^2}{x^3-1}\,\RMd x=
\frac 13\intop_2^3\frac{(x^3-1)'}{x^3-1}\,\RMd x=
\left[\frac 13\ln|x^3-1|\right]_2^3=\frac 13\ln\frac{26}7$
%%
\item $\D\intop_\e^{\e^2}\frac{\RMd x}{x\ln x}=
\intop_\e^{\e^2}\frac{\frac 1x}{\ln x}\,\RMd x=
\intop_\e^{\e^2}\frac{(\ln x)'}{\ln x}\,\RMd x=
\left[\ln|\ln x|\right]_\e^{\e^2}=\ln 2$ 
%%
\item $\D\intop_{0}^{\frac{\pi}2} \frac{\sin x-\cos x}{\sin x+\cos x}\,\RMd x=
-\intop_{0}^{\frac{\pi}2} \frac{(\sin x+\cos x)'}{\sin x+\cos x}\,\RMd x=
-\left[\ln|\sin x+\cos x|\right]_{0}^{\frac{\pi}2}=0$
%%
\item $\D\intop_{0}^{\frac{\pi}4} \frac{\sin x-\cos x}{\sin x+\cos x}\,\RMd x=
-\left[\ln|\sin x+\cos x|\right]_{0}^{\frac{\pi}4}=
-\ln\sqrt{2}+\ln 1=-\frac 12\ln 2$
\end{enumerate}
}
