integrate of sin5x/sec^2x dx

Question

kc_kennylau · Answer

$$\int\frac{\sin5x}{\sec^2x}dx$$

kc_kennylau · Answer

Is this what you mean?

anonymous · Answer

yeah

kc_kennylau · Answer

@ganeshie8 @zepdrix

zepdrix · Answer

You can get the sin5x in terms of sinx and cosx by applying the Angle Addition Formula a billion times.. there must be a better way though... hmm

kc_kennylau · Answer

I'm also looking for a better way lol

aravindg · Answer

"billion times" :O

ganeshie8 · Answer

$\large   \int\frac{\sin5x}{\sec^2x}dx  $


$\large   \frac{1}{2}\int \sin 5x   (1+\cos 2x)dx  $


$\large   \frac{1}{2}\int \sin 5x  dx +    \frac{1}{2}\int \sin (5x)\cos (2x)dx  $


$\large   \frac{1}{2}\int \sin 5x  dx +    \frac{1}{4}\int (\sin(7x) + \sin(3x))dx  $

anonymous · Answer

$$\int\limits \frac{ \sin ^5x }{\sec ^2x }dx=\int\limits \sin ^5x \left( 1-\sin ^2x \right)dx$$
$$\int\limits \sin ^5x~dx-\int\limits \sin ^7x~dx$$
$$\int\limits \sin ^5x~dx=\int\limits \sin ^4x~\sin x~dx$$
$$=\int\limits \left( 1-\cos ^2x \right)^2\sin x~dx=\int\limits \left( 1-2\cos ^2x+\cos ^4x \right)\sin x~dx$$
you can complete now.
$$\int\limits f^n(x)f \prime \left( x \right)dx=\frac{ f ^{n+1}(x) }{n+1}$$

anonymous · Answer

if it is sin 5x then ganeshie8 solution is best