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

The integral of 25sec^5(x) is 25((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 25sec^5(x)

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

25 (\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 25 sec^{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) d x

Apply Integral Reduction

= 25 (\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) ∣

= 25 (\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

= 25 (\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

y^{^{''}} + 7 y^{^{'}} + 10 y = 14 x e^{3 x}

\frac{d u}{d v}, u + v = 5

\int (\frac{1}{3} e^{x + 3} + \frac{3}{x} - 5) d x

\int \frac{x ^{2} - 11 x - 18}{x ( x ^{2} + 3 x + 3 )} d x

\frac{\partial}{\partial x} (5 x y - 4 x^{2} y)

What is the integral of 25sec^5(x) ?
The integral of 25sec^5(x) is 25((sec^4(x)sin(x))/4+3/4 (1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|))+C