What is the integral of 1/(ysqrt(3+(ln(y))2)) ?

The integral of 1/(ysqrt(3+(ln(y))2)) is sqrt(3+2ln(y))+C

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integral of 1/(ysqrt(3+(ln(y))2))

\int \frac{1}{y 3 + ( ln ( y ) ) 2} d y

3 + 2 ln (y) + C

Solution steps

\int \frac{1}{y 3 + ln ( y ) \cdot 2} d y

Apply u-substitution

: \int \frac{1}{3 + 2 u} d u

= \int \frac{1}{3 + 2 u} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= \int \frac{1}{2 v} d v

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}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{2} \cdot 2 (3 + 2 ln (y))^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (3 + 2 ln (y))^{\frac{1}{2}} : 3 + 2 ln (y)

= 3 + 2 ln (y)

Add a constant to the solution

= 3 + 2 ln (y) + C

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What is the integral of 1/(ysqrt(3+(ln(y))2)) ?
The integral of 1/(ysqrt(3+(ln(y))2)) is sqrt(3+2ln(y))+C