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

The integral of 5tan^3(x)sec^3(x) is sec^5(x)-5/3 sec^3(x)+C

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

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

sec^{5} (x) - \frac{5}{3} sec^{3} (x) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 5 \cdot \int (- 1 + sec^{2} (x)) tan (x) sec^{3} (x) d x

Apply u-substitution

: \int u^{2} (u^{2} - 1) d u

= 5 \cdot \int u^{2} (u^{2} - 1) d u

Expand

u^{2} (u^{2} - 1) : u^{4} - u^{2}

= 5 \cdot \int u^{4} - u^{2} d u

Apply the Sum Rule:

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

= 5 (\int u^{4} d u - \int u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{2} d u = \frac{u ^{3}}{3}

= 5 (\frac{u ^{5}}{5} - \frac{u ^{3}}{3})

Substitute back

u = sec (x)

= 5 (\frac{sec ^{5} ( x )}{5} - \frac{sec ^{3} ( x )}{3})

Simplify

5 (\frac{sec ^{5} ( x )}{5} - \frac{sec ^{3} ( x )}{3}) : sec^{5} (x) - \frac{5}{3} sec^{3} (x)

= sec^{5} (x) - \frac{5}{3} sec^{3} (x)

Add a constant to the solution

= sec^{5} (x) - \frac{5}{3} sec^{3} (x) + C

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