What is the integral of tan^5(5x)(sec^3(5x)-1) ?

The integral of tan^5(5x)(sec^3(5x)-1) is

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integral of tan^5(5x)(sec^3(5x)-1)

\int tan^{5} (5 x) (sec^{3} (5 x) - 1) d x

\frac{1}{35} sec^{7} (5 x) - \frac{2}{25} sec^{5} (5 x) + \frac{1}{15} sec^{3} (5 x) - \frac{1}{5} ln ∣ sec (5 x) ∣ + \frac{1}{5} sec^{2} (5 x) - \frac{1}{20} sec^{4} (5 x) + C

Solution steps

\int tan^{5} (5 x) (sec^{3} (5 x) - 1) d x

Expand

tan^{5} (5 x) (sec^{3} (5 x) - 1) : tan^{5} (5 x) sec^{3} (5 x) - tan^{5} (5 x)

= \int tan^{5} (5 x) sec^{3} (5 x) - tan^{5} (5 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int tan^{5} (5 x) sec^{3} (5 x) d x - \int tan^{5} (5 x) d x

\int tan^{5} (5 x) sec^{3} (5 x) d x = \frac{sec ^{7} ( 5 x )}{35} - \frac{2 sec ^{5} ( 5 x )}{25} + \frac{sec ^{3} ( 5 x )}{15}

\int tan^{5} (5 x) d x = \frac{1}{5} ln ∣ sec (5 x) ∣ - \frac{sec ^{2} ( 5 x )}{5} + \frac{sec ^{4} ( 5 x )}{20}

= \frac{sec ^{7} ( 5 x )}{35} - \frac{2 sec ^{5} ( 5 x )}{25} + \frac{sec ^{3} ( 5 x )}{15} - (\frac{1}{5} ln ∣ sec (5 x) ∣ - \frac{sec ^{2} ( 5 x )}{5} + \frac{sec ^{4} ( 5 x )}{20})

Simplify

\frac{sec ^{7} ( 5 x )}{35} - \frac{2 sec ^{5} ( 5 x )}{25} + \frac{sec ^{3} ( 5 x )}{15} - (\frac{1}{5} ln ∣ sec (5 x) ∣ - \frac{sec ^{2} ( 5 x )}{5} + \frac{sec ^{4} ( 5 x )}{20}) : \frac{1}{35} sec^{7} (5 x) - \frac{2}{25} sec^{5} (5 x) + \frac{1}{15} sec^{3} (5 x) - \frac{1}{5} ln ∣ sec (5 x) ∣ + \frac{1}{5} sec^{2} (5 x) - \frac{1}{20} sec^{4} (5 x)

= \frac{1}{35} sec^{7} (5 x) - \frac{2}{25} sec^{5} (5 x) + \frac{1}{15} sec^{3} (5 x) - \frac{1}{5} ln ∣ sec (5 x) ∣ + \frac{1}{5} sec^{2} (5 x) - \frac{1}{20} sec^{4} (5 x)

Add a constant to the solution

= \frac{1}{35} sec^{7} (5 x) - \frac{2}{25} sec^{5} (5 x) + \frac{1}{15} sec^{3} (5 x) - \frac{1}{5} ln ∣ sec (5 x) ∣ + \frac{1}{5} sec^{2} (5 x) - \frac{1}{20} sec^{4} (5 x) + C

\frac{\partial}{\partial x} (y \cdot ln (x^{4} + y^{4} + 1) - 8)

x \to 0 lim (\frac{( 343 + x ) ^{\frac{1}{3}} - 7}{( x )})

\int \frac{5 x + 2}{( x - 5 ) ( x + 5 ) ( x - 1 )} d x

\frac{d}{d x} (\frac{x y ^{3}}{x ^{2} + y ^{2}})

\int x^{2} + 4 x + 5 d x

What is the integral of tan^5(5x)(sec^3(5x)-1) ?
The integral of tan^5(5x)(sec^3(5x)-1) is