What is the integral from 0 to sqrt(x of) y/(x^2+1) ?

The integral from 0 to sqrt(x of) y/(x^2+1) is x/(2(1+x^2))

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integral from 0 to sqrt(x of) y/(x^2+1)

\int_{0}^{x} \frac{y}{x ^{2} + 1} d y

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

Solution steps

\int_{0}^{x} \frac{y}{x ^{2} + 1} d y

Take the constant out:

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

= \frac{1}{1 + x ^{2}} \cdot \int_{0}^{x} y d y

Apply the Power Rule

: [\frac{y ^{2}}{2}]_{0}^{x}

= \frac{1}{1 + x ^{2}} [\frac{y ^{2}}{2}]_{0}^{x}

Simplify

\frac{1}{1 + x ^{2}} [\frac{y ^{2}}{2}]_{0}^{x} : \frac{[ \frac{y ^{2}}{2} ] _{0}^{x}}{1 + x ^{2}}

= \frac{[ \frac{y ^{2}}{2} ] _{0}^{x}}{1 + x ^{2}}

Compute the boundaries:

\frac{x}{2}

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

Simplify

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

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\int_{0}^{2} (2 x - 3) (8 x^{2} + 6) d x

\int_{0}^{2} t e^{- t^{2}} d t

\int_{0}^{3} x^{2} \cdot e^{x} d x

What is the integral from 0 to sqrt(x of) y/(x^2+1) ?
The integral from 0 to sqrt(x of) y/(x^2+1) is x/(2(1+x^2))