Integration by subsititution! (1 Viewer)

[z600]

Can anyone do the following?

1) The derivative of a equation is root(x^2-9)/x and f(x)=3 (let x=3sec theta)

Find the equation

2) find the integral of

1/(4+x^2)^(3/2) (let x=2tan theta)


thanks

 

[pLuvia]

1. Using that substitution

int. (sqrt{9sec²@-9})/(3sec@)*3sec@tan@d@
=int. 3tan²@*tan@ d@
=int. 3(1-sec²@) d@
=3@-tan@+C
=3cos^-1(3/x)-tan(cos^-1(3/x))+C

May have made a mistake somewhere, havent done this in ages

2. Use that substitution again
int. 1/(4+4tan²@)^(3/2)*2sec²@ d@
=int. 1/(4sec²@)^(3/2)*2sec²@ d@
=int. 1/(8sec³@)*2sec²@ d@
=int. 1/4sec@ d@
=int. cos@/4 d@
=1/4sin@+C
=1/4sin(tan^-1(x/2))+C

 

[RedZenith]

Meh... I got 3sec^{-1}(x/3)-\sqrt{x^2-3}+C for the first one