What is the integral of 2tan(x)sec^5(x) ?

The integral of 2tan(x)sec^5(x) is 2/5 sec^5(x)+C

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integral of 2tan(x)sec^5(x)

\int 2 tan (x) sec^{5} (x) d x

\frac{2}{5} sec^{5} (x) + C

Solution steps

\int 2 tan (x) sec^{5} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int tan (x) sec^{5} (x) d x

Apply u-substitution

: \int u^{4} d u

= 2 \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= 2 \cdot \frac{u ^{5}}{5}

Substitute back

u = sec (x)

= 2 \cdot \frac{sec ^{5} ( x )}{5}

Simplify

2 \cdot \frac{sec ^{5} ( x )}{5} : \frac{2}{5} sec^{5} (x)

= \frac{2}{5} sec^{5} (x)

Add a constant to the solution

= \frac{2}{5} sec^{5} (x) + C

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What is the integral of 2tan(x)sec^5(x) ?
The integral of 2tan(x)sec^5(x) is 2/5 sec^5(x)+C