Question

4. a) Evaluate : integrate (x ^ 3 + 1/(x ^ 3)) dx

b) int 0 ^ pi 2 -4 sin xdx 3+4 cos x

Answer 1

4.

a)

$$\int (x^3+\dfrac{1}{x^3})dx=\dfrac{x^4}{4}-\dfrac{1}{2x^2}+C$$

b)

$$\displaystyle\int_{0}^{\pi/2}\dfrac{-4\sin x}{3+4\cos x}dx$$

$$\int \dfrac{-4\sin x}{3+4\cos x}dx$$

Use $u$ -substitution

$u=3+4\cos x, du=-4\sin x$

$$\int \dfrac{-4\sin x}{3+4\cos x}dx=\int \dfrac{du}{u}dx=\ln(|u|)+C$$

$$=\ln(|3+4\cos x|)+C$$

$$\displaystyle\int_{0}^{\pi/2}\dfrac{-4\sin x}{3+4\cos x}dx=[\ln(|3+4\cos x|)]\begin{matrix} \pi/2 \\ 0 \end{matrix}$$

$$=\ln(|3+4\cos (\pi/2)|)-\ln(|3+4\cos (0)|)$$

$$=\ln 3-\ln 7=\ln (3/7)$$