What is the integral from 0 to x of 3/(x^2+y^2) ?

The integral from 0 to x of 3/(x^2+y^2) is (3pi)/(4x)

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integral from 0 to x of 3/(x^2+y^2)

\int_{0}^{x} \frac{3}{x ^{2} + y ^{2}} d y

\frac{3 π}{4 x}

Solution steps

\int_{0}^{x} \frac{3}{x ^{2} + y ^{2}} d y

Take the constant out:

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

= 3 \cdot \int_{0}^{x} \frac{1}{x ^{2} + y ^{2}} d y

Apply Integral Substitution

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

Take the constant out:

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

= 3 \cdot \frac{1}{x} \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 3 \cdot \frac{1}{x} [arctan (u)]_{0}^{1}

Simplify

3 \cdot \frac{1}{x} [arctan (u)]_{0}^{1} : \frac{3}{x} [arctan (u)]_{0}^{1}

= \frac{3}{x} [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{3}{x} \cdot \frac{π}{4}

Simplify

= \frac{3 π}{4 x}

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What is the integral from 0 to x of 3/(x^2+y^2) ?
The integral from 0 to x of 3/(x^2+y^2) is (3pi)/(4x)