What is the integral of 1/(ysqrt(9y^2-16)) ?

The integral of 1/(ysqrt(9y^2-16)) is 1/4 arcsec(3/4 y)+C

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integral of 1/(ysqrt(9y^2-16))

\int \frac{1}{y 9 y ^{2} - 16} d y

\frac{1}{4} arcsec (\frac{3}{4} y) + C

Solution steps

\int \frac{1}{y 9 y ^{2} - 16} d y

Apply Trigonometric Substitution

= \int \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{1}{4} \cdot 1 \cdot u

Substitute back

u = arcsec (\frac{3}{4} y)

= \frac{1}{4} \cdot 1 \cdot arcsec (\frac{3}{4} y)

Simplify

= \frac{1}{4} arcsec (\frac{3}{4} y)

Add a constant to the solution

= \frac{1}{4} arcsec (\frac{3}{4} y) + C

What is the integral of 1/(ysqrt(9y^2-16)) ?
The integral of 1/(ysqrt(9y^2-16)) is 1/4 arcsec(3/4 y)+C