What is the integral of 3/(ysqrt(4-(ln(y))^2)) ?

The integral of 3/(ysqrt(4-(ln(y))^2)) is 3arcsin(1/2 ln(y))+C

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integral of 3/(ysqrt(4-(ln(y))^2))

\int \frac{3}{y 4 - ( ln ( y ) ) ^{2}} d y

3 arcsin (\frac{1}{2} ln (y)) + C

Solution steps

\int \frac{3}{y 4 - ( ln ( 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 \frac{1}{y 4 - ( ln ( y ) ) ^{2}} d y

Simplify

= 3 \cdot \int \frac{1}{y 4 - ln ^{2} ( y )} d y

Apply u-substitution

: \int \frac{1}{4 - u ^{2}} d u

= 3 \cdot \int \frac{1}{4 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 3 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 3 \cdot 1 \cdot v

Substitute back

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

Simplify

= 3 arcsin (\frac{1}{2} ln (y))

Add a constant to the solution

= 3 arcsin (\frac{1}{2} ln (y)) + C

n = 1 \sum \infty \frac{1}{( n + 7 ) n}

f (x) = x - 2, (6, 2)

\frac{d y}{d x} 7 + e^{- 5 x} y = 0

f (x) = arctan (x^{2})

(ln (x^{2} - 10 x - 10)), at x = 11

What is the integral of 3/(ysqrt(4-(ln(y))^2)) ?
The integral of 3/(ysqrt(4-(ln(y))^2)) is 3arcsin(1/2 ln(y))+C