compute-sec-5-x-tan-3-x-dx-

Question Number 12533 by tawa last updated on 24/Apr/17

compute

\int \sec^{5} (x) \tan^{3} (x) dx

Answered by sma3l2996 last updated on 25/Apr/17

I = \int {s e c}^{5} (x) {t a n}^{3} (x) d x = \int \frac{{s i n}^{3} x}{{c o s}^{8} x} d x

u = {s i n}^{2} x \Rightarrow u^{'} = 2 c o s x s i n x

v^{'} = \frac{s i n x}{{c o s}^{8} x} \Rightarrow v = \frac{1}{7 {c o s}^{7} x}

I = \frac{{s i n}^{2} x}{7 {c o s}^{7} x} - \frac{2}{7} \int \frac{s i n x}{{c o s}^{6} x} d x + c

I = \frac{{s i n}^{2} x}{7 {c o s}^{7} x} - \frac{2}{7} \times (\frac{1}{5 {c o s}^{5} x}) + C

= \frac{1}{7} {s e c}^{5} (x) {t a n}^{2} (x) - \frac{2}{35} {s e c}^{5} (x) + C

I = \frac{1}{7} {s e c}^{5} (x) ({t a n}^{2} (x) - \frac{2}{5}) + C

Commented by tawa last updated on 25/Apr/17

God bless you sir .

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