Question

hello guys i confuse how to clear this question.
can someone help me?
1) integral sec^4 (2x) dx
2) sin^3 (3x) cos^5(3x) dx

Answer 1

\[\int\limits \sec ^4(2x) dx=\int\limits \sec ^2(2x) \sec ^2(2x) dx=\int\limits \left( 1+\tan ^2(2x) \right)\sec ^2(2x)dx\]
\[=\int\limits \sec ^2(2x)dx+\int\limits \tan ^2(2x)\sec ^2(2x)dx+c\]
\[\int\limits f^n \left( x \right)f \prime \left( x \right)dx=\frac{ f ^{n+1}\left( x \right) }{ n+1 }\]
in the second put tan 2x=t,diff.

Answer 2

2.\[\int\limits \sin ^3 x \cos ^5x~dx=\int\limits \cos ^5x \sin ^2x \sin x~dx\]
\[=\int\limits \cos ^5x~\left( 1-\cos ^2x \right)\sin x~dx=\int\limits \cos ^5x ~\sin x~dx-\int\limits \cos ^7x~\sin x~dx+c\]
solve it.