What is the integral of 25sec(5x)tan(5x)sec^4(5x) ?

The integral of 25sec(5x)tan(5x)sec^4(5x) is sec^5(5x)+C

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integral of 25sec(5x)tan(5x)sec^4(5x)

\int 25 sec (5 x) tan (5 x) sec^{4} (5 x) d x

sec^{5} (5 x) + C

Solution steps

\int 25 sec (5 x) tan (5 x) sec^{4} (5 x) d x

Take the constant out:

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

= 25 \cdot \int sec (5 x) tan (5 x) sec^{4} (5 x) d x

Simplify

sec (5 x) tan (5 x) sec^{4} (5 x) : sec^{5} (5 x) tan (5 x)

= 25 \cdot \int sec^{5} (5 x) tan (5 x) d x

Apply u-substitution

: \int sec^{5} (u) tan (u) \frac{1}{5} d u

= 25 \cdot \int sec^{5} (u) tan (u) \frac{1}{5} d u

Take the constant out:

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

= 25 \cdot \frac{1}{5} \cdot \int sec^{5} (u) tan (u) d u

Apply u-substitution

: \int v^{4} d v

= 25 \cdot \frac{1}{5} \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= 25 \cdot \frac{1}{5} \cdot \frac{v ^{5}}{5}

Substitute back

= 25 \cdot \frac{1}{5} \cdot \frac{sec ^{5} ( 5 x )}{5}

Simplify

25 \cdot \frac{1}{5} \cdot \frac{sec ^{5} ( 5 x )}{5} : sec^{5} (5 x)

= sec^{5} (5 x)

Add a constant to the solution

= sec^{5} (5 x) + C

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What is the integral of 25sec(5x)tan(5x)sec^4(5x) ?
The integral of 25sec(5x)tan(5x)sec^4(5x) is sec^5(5x)+C