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Popular >

integral of y/(y^2+z^2)

Solution

\int \frac{y}{y ^{2} + z ^{2}} d y

Solution

\frac{1}{2} ln z^{2} + y^{2} + C

Solution steps

\int \frac{y}{y ^{2} + z ^{2}} d y

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = z^{2} + y^{2}

= \frac{1}{2} ln z^{2} + y^{2}

Add a constant to the solution

= \frac{1}{2} ln z^{2} + y^{2} + C

Popular Examples

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\int sin (π x) - 3 sin (3 x) d x

integral of 1/(7x-4)

\int \frac{1}{7 x - 4} d x

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\int \frac{1}{1 - 1 - x ^{2}} d x

integral of cot^4(x/6)

\int cot^{4} (\frac{x}{6}) d x

integral of 1/((1-sin^2(x))^{3/2)}

\int \frac{1}{( 1 - sin ^{2} ( x ) ) ^{\frac{3}{2}}} d x

Frequently Asked Questions (FAQ)

What is the integral of y/(y^2+z^2) ?
The integral of y/(y^2+z^2) is 1/2 ln|z^2+y^2|+C