How do you integrate ${tan}^{5} (x)$ $tan^5(x)$ ?

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How do you integrate ${tan}^{5} (x)$ $tan^5(x)$ ?

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Answer 1 · 2017-10-31T16:49:05

Recursively use the reduction formula:

$\int {tan}^{n} (x) d x = \frac{1}{n - 1} {tan}^{n - 1} (x) - \int {tan}^{n - 2} (x) d x$ $inttan^n(x)dx=1/(n-1)tan^(n-1)(x)-inttan^(n-2)(x)dx$

Until $n = 1$ $n = 1$ , then we know this one.

Explanation:

Starting with $n = 5$ $n = 5$ :

$\int {tan}^{5} (x) d x = \frac{1}{4} {tan}^{4} (x) - \int {tan}^{3} (x) d x$ $inttan^5(x)dx=1/4tan^4(x)-inttan^3(x)dx$

Now $n = 3$ $n = 3$ :

$\int {tan}^{5} (x) d x = \frac{1}{4} {tan}^{4} (x) - \frac{1}{2} {tan}^{2} (x) + \int tan (x) d x$ $inttan^5(x)dx=1/4tan^4(x)-1/2tan^2(x)+inttan(x)dx$

We know the last integral:

$\int {tan}^{5} (x) d x = \frac{1}{4} {tan}^{4} (x) - \frac{1}{2} {tan}^{2} (x) + ln | sec (x) | + C$ $inttan^5(x)dx=1/4tan^4(x)-1/2tan^2(x)+ln|sec(x)|+ C$

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How do you integrate ${tan}^{5} (x)$ $tan^5(x)$ ?

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