C4 Integration by Substitution

Trying to do this question but getting stuck, tried replacing the dx by dϑ/cosϑ and I still can't see how it all fits together.

The mark scheme says you end up with tanϑ from integrating sec^2ϑ, but i can't see how you even get there in the first place.

Help appreciated!

trig question.JPG8.6KB

Reply 1

Lover's Knot

x = sin(t), dx = cos(t).dt

INT dx/(1-x^2)^(3/2)
= INT cos(t).dt/(1-sin^2(t))^(3/2)
= INT cos(t).dt/(cos^2(t))^(3/2)
= INT cos(t).dt/(cos^3(t))
= INT dt/(cos^2(t))
= INT sec^2(t).dt

Reply 2

ArchedEdge

Lover's Knot

x = sin(t), dx = cos(t).dt

INT dx/(1-x^2)^(3/2)
= INT cos(t).dt/(1-sin^2(t))^(3/2)
= INT cos(t).dt/(cos^2(t))^(3/2)
= INT cos(t).dt/(cos^3(t))
= INT dt/(cos^2(t))
= INT sec^2(t).dt