Question

Evaluate the integral of cos2x dx/(2+5 sin 2x)⁴

Answer 1

$$\int\dfrac{\cos(2x)}{(2+5\sin(2x))^4}dx$$

Use $u$ -substitution

$$u=2+5\sin(2x), du=10\cos(2x)dx$$

$$\int\dfrac{\cos(2x)}{(2+5\sin(2x))^4}dx=\dfrac{1}{10}\int\dfrac{du}{u^4}dx$$

$$=-\dfrac{1}{30u^3}+C=-\dfrac{1}{30(2+5\sin(2x))^3}+C$$

$$\int\dfrac{\cos(2x)}{(2+5\sin(2x))^4}dx=-\dfrac{1}{30(2+5\sin(2x))^3}+C$$