\( x \frac { 1 } { \sqrt { 1 - y ^ { 2 } } } \cdot y ^ { \prime } + 3 x ^ { 2 } = \frac { 1 } { 2 \sqrt { y } } \cdot y ^ { \prime } \)

\( \frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 } \)

\( \iint_{R} \sqrt{x^{2}+y^{2}} d x d y \) बत्र मान निर्

9(a) Find \( \int \frac{x+2}{x^{2}+2 x-3} \mathrm{~d} x \) \( \begin{array}{ll}\text { (b) Find } \int 2 x \sin x \mathrm{~d} x\end{array} \) \( \begin{array}{ll}\text { (c) Using the substitution } x=3 \sin \theta \text {, find the exact value of } \int_{0}^{\frac{3}{2}} \sqrt{9-x^{2}} \mathrm{~d} x \\ 9 & \text { (a) } \frac{3}{4} \ln |x-1|+\frac{1}{4} \ln |x+3|+C \text { or } \frac{1}{2} \ln \left|x^{2}+2 x-3\right|+\frac{1}{4} \ln \left|\frac{x-1}{x+3}\right|+C \\ \text { (b) }-2 x \cos x+2 \sin x+C & \text { (c) } \frac{9}{8} \sqrt{3}+\frac{3}{4} \pi\end{array} \) \( \begin{array}{ll}\text { (b) }\end{array} \)

9. Deducir la fórmula aproximada \[ \sqrt[3]{x+h} \approx \sqrt[3]{x}+\frac{h}{3 \sqrt[3]{x^{2}}} \] y halla los valores aproximados de \( \sqrt[3]{10} \) y \( \sqrt[3]{27.3} \)

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal manamum value at \( x=1 \), + What is the value of a \( f c \) ?