What is the integral of ((sin(x)+1)^{3/2})/(sec(x)) ?

The integral of ((sin(x)+1)^{3/2})/(sec(x)) is 2/5 (sin(x)+1)^{5/2}+C

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integral of ((sin(x)+1)^{3/2})/(sec(x))

\int \frac{( sin ( x ) + 1 ) ^{\frac{3}{2}}}{sec ( x )} d x

\frac{2}{5} (sin (x) + 1)^{\frac{5}{2}} + C

Solution steps

\int \frac{( sin ( x ) + 1 ) ^{\frac{3}{2}}}{sec ( x )} d x

Apply u-substitution

: \int (u + 1)^{\frac{3}{2}} d u

= \int (u + 1)^{\frac{3}{2}} d u

Apply u-substitution

: \int v^{\frac{3}{2}} d v

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

Apply the Power Rule

: \frac{2}{5} v^{\frac{5}{2}}

= \frac{2}{5} v^{\frac{5}{2}}

Substitute back

= \frac{2}{5} (sin (x) + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (sin (x) + 1)^{\frac{5}{2}} + C

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

f (x) = 6 x^{2} + 2 x - 6

\frac{d}{d x} (\frac{x + 1}{x - 3})

y = arcsin (6 x + 1)

\frac{d ^{3}}{d x ^{3}} (3 n^{- \frac{1}{3}} + 2 n^{\frac{5}{3}})

What is the integral of ((sin(x)+1)^{3/2})/(sec(x)) ?
The integral of ((sin(x)+1)^{3/2})/(sec(x)) is 2/5 (sin(x)+1)^{5/2}+C