What is the integral of sin(arctan(x)) ?

The integral of sin(arctan(x)) is sqrt(1+x^2)+C

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integral of sin(arctan(x))

\int sin (arctan (x)) d x

1 + x^{2} + C

Solution steps

\int sin (arctan (x)) d x

Simplify

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

= \int \frac{x}{1 + x ^{2}} d x

Apply u-substitution

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

= \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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + x^{2}

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

Simplify

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

= 1 + x^{2}

Add a constant to the solution

= 1 + x^{2} + C

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What is the integral of sin(arctan(x)) ?
The integral of sin(arctan(x)) is sqrt(1+x^2)+C