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Popular >

integral of y/(3y^2+1)

Solution

\int \frac{y}{3 y ^{2} + 1} d y

Solution

\frac{1}{6} ln 3 y^{2} + 1 + C

Solution steps

\int \frac{y}{3 y ^{2} + 1} d y

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = 3 y^{2} + 1

= \frac{1}{6} ln 3 y^{2} + 1

Add a constant to the solution

= \frac{1}{6} ln 3 y^{2} + 1 + C

Popular Examples

integral of 6/(7+y^2)

\int \frac{6}{7 + y ^{2}} d y

integral of y/(x(2-x^2))

\int \frac{y}{x ( 2 - x ^{2} )} d x

integral of (4x)/(sqrt(2x^2+3))

\int \frac{4 x}{2 x ^{2} + 3} d x

integral of x/(x^2+6x+8)

\int \frac{x}{x ^{2} + 6 x + 8} d x

integral of 1/(sqrt(16x^2-121))

\int \frac{1}{16 x ^{2} - 121} d x

Frequently Asked Questions (FAQ)

What is the integral of y/(3y^2+1) ?
The integral of y/(3y^2+1) is 1/6 ln|3y^2+1|+C