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

The integral of 5sec^5(x) is 5((sec^4(x)sin(x))/4+3/4 (1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|))+C

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integral of 5sec^5(x)

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

5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) + C

Solution steps

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

Take the constant out:

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

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

Apply Integral Reduction

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} \cdot \int sec^{3} (x) d x)

\int sec^{3} (x) d x = \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣))

Add a constant to the solution

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) + C

\int - e^{- t} + 1 d t

f (x) = x^{4} - 10 x^{2} + 9, at x = 1

h (g (x) k (x))

\frac{2}{3} (1 + x)^{\frac{3}{2}}

f (x) = 3 e^{x}, at x = 0

What is the integral of 5sec^5(x) ?
The integral of 5sec^5(x) is 5((sec^4(x)sin(x))/4+3/4 (1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|))+C