Integration by parts (?) with integrals

[donnac10]

Hi everybody!

I'm new here and I really need help with my homework... My problem is:

tan3(5x)sec(5x)dx ​

My first initial thought when I saw this is to take out one tan so it would be

tan^2(5x)tan(5x)sec(5x)dx

then it would be,

(1-sec^2(5x))tan(5x)sec(5x)dx

This is where I got stuck... I don't know what to do after this..

Please help me! Thanks so much for all y'all!

 

[galactus]

Parts is not necessary. Split off a \(\displaystyle sec(5x)tan(5x)\) and note that \(\displaystyle tan^{2}(5x)=sec^{2}(5x)-1\)

This gives \(\displaystyle sec(5x)tan(5x)(sec^{2}(5x)-1)dx\)

Now, make the sub \(\displaystyle u=sec(5x), \;\ du/5=sec(5x)tan(5x)dx\)

 

[donnac10]

Ohh.. That makes a lot of sense! Thank you so much for your help!

Parts is not necessary. Split off a \(\displaystyle sec(5x)tan(5x)\) and note that \(\displaystyle tan^{2}(5x)=sec^{2}(5x)-1\)

This gives \(\displaystyle sec(5x)tan(5x)(sec^{2}(5x)-1)dx\)

Now, make the sub \(\displaystyle u=sec(5x), \;\ du/5=sec(5x)tan(5x)dx\)