Integrate the given function. \[ \int \frac{6+e^{x}}{6 x+e^{x}} d x \] \( \int \frac{6+e^{x}}{6 x+e^{x}} d x=\square \) (Type an exact answer.)

Use the substitution \( u=x^{2}+7 \) to find the following indefinite integral. Check your answer by differentiation. \[ \int 2 x\left(x^{2}+7\right)^{4} d x \] \( \int 2 x\left(x^{2}+7\right)^{4} d x=\frac{\left(x^{2}+7\right)^{5}}{5}+C \)

Q. \( 10[\mathrm{CO} 4] \) Find the maxima and minima of the following functions, if exist: \( \begin{array}{l}\text { a) } f(x)=x^{3}-3 x^{2}+2 \\ \text { b) } f(x)=x^{2}+1 \\ \text { c) } f(x)=3 x^{3}-9 x+1\end{array} \)

\( \lim _ { n i n } \frac { ( n + 1 ) ^ { 3 } - ( n + 2 ) ^ { 2 } } { ( n - 2 ) ^ { 3 } - ( n + 3 ) ^ { 3 } } \)

3. Use the value of the first integral \( I \) to evaluate the two given integrals. \( I=\int_{0}^{3}\left(x^{3}-4 x\right) d x=\frac{9}{4} \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x=\frac{-27}{2} \) (Simplify your answer.) \( \int_{3}^{0}\left(4 x-x^{3}\right) d x \) b. \( \int_{3}^{0}\left(4 x-x^{3}\right) d x=\square \frac{9}{4} \) (Simplify your answer.)

10. Evaluate the following integral. \( \int \frac{x^{2}}{x^{3}+2} d x \) \( \left.\int \frac{x^{2}}{x^{3}+2} d x=\frac{1}{3}|n| x+2 \right\rvert\,+C \)