Answer to Question #195416 in Civil and Environmental Engineering for John Prats

Question

Evaluate the integral cos⁷3x dx with the limits from 0 to π/6.

Answer 1

$\int _0^{\frac{\pi }{2}}\cos ^7\left(3x\right)\frac{1}{3}d3x$

$=\frac{1}{3}\cdot \int _0^{\frac{\pi }{2}}\cos ^7\left(3x\right)d3x$

$=\frac{1}{3}\cdot \int _0^{\frac{\pi }{2}}\left(1-\sin ^2\left(3x\right)\right)^3\cos \left(3x\right)d3x$

$=\frac{1}{3}\cdot \int _0^1\left(1-v^2\right)^3dv$

$=\frac{1}{3}\cdot \int _0^11-3v^2+3v^4-v^6dv$

$=\frac{1}{3}\left(\int _0^11dv-\int _0^13v^2dv+\int _0^13v^4dv-\int _0^1v^6dv\right)$

$=\frac{1}{3}\left(1-1+\frac{3}{5}-\frac{1}{7}\right)$

$=\frac{16}{105}$