What is the integral of sqrt(1+tan(h^2)x) ?

The integral of sqrt(1+tan(h^2)x) is cot(h^2) 2/3 (1+tan(h^2)x)^{3/2}+C

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integral of sqrt(1+tan(h^2)x)

\int 1 + tan (h^{2}) x d x

cot (h^{2}) \frac{2}{3} (1 + tan (h^{2}) x)^{\frac{3}{2}} + C

Solution steps

\int 1 + tan (h^{2}) x d x

Apply u-substitution

= \int u cot (h^{2}) d u

Take the constant out:

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

= cot (h^{2}) \cdot \int u d u

Apply the Power Rule

= cot (h^{2}) \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + tan (h^{2}) x

= cot (h^{2}) \frac{2}{3} (1 + tan (h^{2}) x)^{\frac{3}{2}}

Add a constant to the solution

= cot (h^{2}) \frac{2}{3} (1 + tan (h^{2}) x)^{\frac{3}{2}} + C

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