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# C4 Integration by Substitution watch

1. 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!
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2. 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
3. (Original post by 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
Ahh thanks for that. I was being stupid with the whole dϑ thing. cheers!

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