∫xsec^3 xdx please help


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Question Number 60346 by necx1 last updated on 20/May/19

$\int x \sec^{3} x d x$ $p l e a s e h e l p$
Commented by kaivan.ahmadi last updated on 20/May/19

$f i r s t \int {s e c}^{3} x d x = (\frac{t g x}{c o s x} + l n ∣ s e c x + t g x ∣) / 2$ $n o w l e t$ $u = x \Rightarrow d u = d x$ $d v = {s e c}^{3} x d x \Rightarrow v = (\frac{t g x}{c o s x} + l n ∣ s e c x + t g x ∣) / 2$ $u v - \int v d u = x (\frac{t g x}{c o s x} + l b ∣ s e c x + t g x ∣) / 2 - \frac{1}{2} \int (\frac{t g x}{c o s x} + l n ∣ s e c x + t g x ∣) d x$ $o n t h e o t h e r h a n d$ $\int \frac{t g x}{c o s x} d x = \int \frac{s i n x}{{c o s}^{2} x} d x = \frac{1}{c o s x} = s e c x$ $a n d$ $\int l n ∣ s e c x + t g x ∣ d x = x l n (s e c x + t g x) - s e c x - t g x$
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