What is the integral of (sqrt(8+tan(x)))/(1-sin^2(x)) ?

The integral of (sqrt(8+tan(x)))/(1-sin^2(x)) is 2/3 (8+tan(x))^{3/2}+C

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integral of (sqrt(8+tan(x)))/(1-sin^2(x))

\int \frac{8 + tan ( x )}{1 - sin ^{2} ( x )} d x

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

Solution steps

\int \frac{8 + tan ( x )}{1 - sin ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{8 + tan ( x )}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{2} (x) 8 + tan (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 + tan (x)

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

Add a constant to the solution

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

\int \frac{4}{t ^{2}} d t

22 x^{3}

\frac{\partial}{\partial x} (y + (cos^{2} (x)))

x \to 2 lim (sin (x))

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

What is the integral of (sqrt(8+tan(x)))/(1-sin^2(x)) ?
The integral of (sqrt(8+tan(x)))/(1-sin^2(x)) is 2/3 (8+tan(x))^{3/2}+C