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

The integral of x/((1+x^2)^{1/2)} is (1+x^2)^{1/2}+C

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integral of x/((1+x^2)^{1/2)}

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Apply u-substitution

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

= \int \frac{1}{2 u ^{\frac{1}{2}}} 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 ^{\frac{1}{2}}} 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})^{\frac{1}{2}}

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

Add a constant to the solution

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

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\int_{0}^{1} \frac{1}{( x ^{2} + 1 ) ^{3}} d x

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\frac{2}{( s ^{2} - 1 ) ( s - 1 )}

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