What is the integral of 1/(xsqrt(225x^2-225)) ?

The integral of 1/(xsqrt(225x^2-225)) is 1/15 arctan(sqrt(x^2-1))+C

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integral of 1/(xsqrt(225x^2-225))

\int \frac{1}{x 225 x ^{2} - 225} d x

\frac{1}{15} arctan (x^{2} - 1) + C

Solution steps

\int \frac{1}{x 225 x ^{2} - 225} d x

Apply Integration By Parts

: \frac{1}{15 x ^{2} - 1} - \int - \frac{2 x ^{2} - 1}{15 x ( x ^{2} - 1 ) x ^{2} - 1} d x

= \frac{1}{15 x ^{2} - 1} - \int - \frac{2 x ^{2} - 1}{15 x ( x ^{2} - 1 ) x ^{2} - 1} d x

\int - \frac{2 x ^{2} - 1}{15 x ( x ^{2} - 1 ) x ^{2} - 1} d x = - \frac{1}{15} (- \frac{1}{x ^{2} - 1} + arctan (x^{2} - 1))

= \frac{1}{15 x ^{2} - 1} - (- \frac{1}{15} (- \frac{1}{x ^{2} - 1} + arctan (x^{2} - 1)))

Simplify

\frac{1}{15 x ^{2} - 1} - (- \frac{1}{15} (- \frac{1}{x ^{2} - 1} + arctan (x^{2} - 1))) : \frac{1}{15} arctan (x^{2} - 1)

= \frac{1}{15} arctan (x^{2} - 1)

Add a constant to the solution

= \frac{1}{15} arctan (x^{2} - 1) + C

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What is the integral of 1/(xsqrt(225x^2-225)) ?
The integral of 1/(xsqrt(225x^2-225)) is 1/15 arctan(sqrt(x^2-1))+C