What is the integral of (arccos(5x))/(sqrt(1-25x^2)) ?

The integral of (arccos(5x))/(sqrt(1-25x^2)) is 1/5 arccos(5x)arcsin(5x)+1/10 arcsin^2(5x)+C

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integral of (arccos(5x))/(sqrt(1-25x^2))

\int \frac{arccos ( 5 x )}{1 - 25 x ^{2}} d x

\frac{1}{5} arccos (5 x) arcsin (5 x) + \frac{1}{10} arcsin^{2} (5 x) + C

Solution steps

\int \frac{arccos ( 5 x )}{1 - 25 x ^{2}} d x

Apply Integration By Parts

: \frac{1}{5} arccos (5 x) arcsin (5 x) - \int - \frac{arcsin ( 5 x )}{1 - 25 x ^{2}} d x

= \frac{1}{5} arccos (5 x) arcsin (5 x) - \int - \frac{arcsin ( 5 x )}{1 - 25 x ^{2}} d x

\int - \frac{arcsin ( 5 x )}{1 - 25 x ^{2}} d x = - \frac{1}{10} arcsin^{2} (5 x)

= \frac{1}{5} arccos (5 x) arcsin (5 x) - (- \frac{1}{10} arcsin^{2} (5 x))

Simplify

= \frac{1}{5} arccos (5 x) arcsin (5 x) + \frac{1}{10} arcsin^{2} (5 x)

Add a constant to the solution

= \frac{1}{5} arccos (5 x) arcsin (5 x) + \frac{1}{10} arcsin^{2} (5 x) + C

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What is the integral of (arccos(5x))/(sqrt(1-25x^2)) ?
The integral of (arccos(5x))/(sqrt(1-25x^2)) is 1/5 arccos(5x)arcsin(5x)+1/10 arcsin^2(5x)+C