1. ## tan sec identity

Hi so the problem is to evaluate

integral sign ("S") tan^4 y sec^3 y dy

now the identity says if the power of tan is odd - which it isn't - or if the power of sec is even which it isn't so what do I do with this so I can use the identity. I thought about (tan^2 y)^2 sec^2 y sec y dy but I think I am just going to end up with leftovers I don't know what to do with

Direction is appreciated.

Thanks
Calculus Beginner

2. Originally Posted by calcbeg
Hi so the problem is to evaluate

integral sign ("S") tan^4 y sec^3 y dy

now the identity says if the power of tan is odd - which it isn't - or if the power of sec is even which it isn't so what do I do with this so I can use the identity. I thought about (tan^2 y)^2 sec^2 y sec y dy but I think I am just going to end up with leftovers I don't know what to do with

Direction is appreciated.

Thanks
Calculus Beginner
$\int \tan ^4 y \sec ^3 y dy$

$\tan ^2 y = \sec ^2 y -1$

$\int (\sec ^2y -1 )^2 \sec ^3 y dy$

$\int \sec ^7 y -2\sec ^5 y + \sec ^3y\cdot dy$

in general

$\int \sec ^n y\cdot dy = \frac{1}{n-1} \sec ^{n-2} y \tan y + \frac{n-2}{n-1} \int \sec ^{n-2} y\cdot dy , 2\leq n$

apply it more than one time if n is more than 3 in the end you will get

$\int \sec y \cdot dy$ and this equal

$\int \sec y \left( \frac{\sec y + \tan y }{\sec y + \tan y }\right) \cdot dy$

$\int \frac{\sec ^2y + \sec y \tan y }{\sec y + \tan y } \cdot dy$

the numerator is the derivative of the denominator so the result is ln(denominator )

$\int \frac{\sec ^2y + \sec y \tan y }{\sec y + \tan y } \cdot dy= \ln (\sec y + \tan y ) + c$