What is $\int {tan}^{3} (x) \cdot sec (x) d x$ $int tan^3(x)*sec(x)dx$ ?

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What is $\int {tan}^{3} (x) \cdot sec (x) d x$ $int tan^3(x)*sec(x)dx$ ?

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Answer 1 · 2017-04-13T00:39:35

$= \frac{{sec}^{3} x}{3} - sec x + C$ $=(sec^3 x)/3-sec x+C$

Explanation:

By rewriting a bit,

$\int {tan}^{3} x sec x d x = \int {tan}^{2} x \cdot sec x tan x d x = \int ({sec}^{2} x - 1) sec x tan x d x$ $int tan^3 x sec x dx=int tan^2 x cdot secx tan x dx=int(sec^2 x-1)sec x tan x dx$

By substitution $u = sec x$ $u=sec x$ , $\Rightarrow d u = sec x tan x d x$ $Rightarrow du = sec x tan x dx$ ,

$= \int (u^{2} - 1) d u = \frac{u^{3}}{3} - u + C = \frac{{sec}^{3} x}{3} - sec x + C$ $=int(u^2-1)du =u^3/3-u+C =(sec^3 x)/3-sec x+C$

I hope that this was clear.

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