What is the integral of (sin(h^{-1})x)/(sqrt(1+x^2)) ?

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

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integral of (sin(h^{-1})x)/(sqrt(1+x^2))

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

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

Solution steps

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

Take the constant out:

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

= sin (h^{- 1}) \cdot \int \frac{x}{1 + x ^{2}} d x

h^{- 1} = \frac{1}{h}

= sin (\frac{1}{h}) \cdot \int \frac{x}{1 + x ^{2}} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= sin (\frac{1}{h}) \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= sin (\frac{1}{h}) \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= sin (\frac{1}{h}) \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + x^{2}

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

Simplify

sin (\frac{1}{h}) \frac{1}{2} \cdot 2 (1 + x^{2})^{\frac{1}{2}} : sin (\frac{1}{h}) 1 + x^{2}

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

Add a constant to the solution

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

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