What is the integral of sec(x)tan(x)sqrt(8+sec(x)) ?

The integral of sec(x)tan(x)sqrt(8+sec(x)) is 2/3 (8+sec(x))^{3/2}+C

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integral of sec(x)tan(x)sqrt(8+sec(x))

\int sec (x) tan (x) 8 + sec (x) d x

\frac{2}{3} (8 + sec (x))^{\frac{3}{2}} + C

Solution steps

\int sec (x) tan (x) 8 + sec (x) d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 8 + sec (x)

= \frac{2}{3} (8 + sec (x))^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (8 + sec (x))^{\frac{3}{2}} + C

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What is the integral of sec(x)tan(x)sqrt(8+sec(x)) ?
The integral of sec(x)tan(x)sqrt(8+sec(x)) is 2/3 (8+sec(x))^{3/2}+C