What is the integral of (6-72x)/(sqrt(25-36x^2)) ?

The integral of (6-72x)/(sqrt(25-36x^2)) is 2sqrt(-36x^2+25)+arcsin(6/5 x)+C

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integral of (6-72x)/(sqrt(25-36x^2))

\int \frac{6 - 72 x}{25 - 36 x ^{2}} d x

2 - 36 x^{2} + 25 + arcsin (\frac{6}{5} x) + C

Solution steps

\int \frac{6 - 72 x}{25 - 36 x ^{2}} d x

Apply Trigonometric Substitution

= \int - 10 sin (u) + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 10 sin (u) d u + \int 1 d u

\int 10 sin (u) d u = - 10 cos (u)

\int 1 d u = u

= - (- 10 cos (u)) + u

Substitute back

u = arcsin (\frac{6}{5} x)

= - (- 10 cos (arcsin (\frac{6}{5} x))) + arcsin (\frac{6}{5} x)

Simplify

- (- 10 cos (arcsin (\frac{6}{5} x))) + arcsin (\frac{6}{5} x) : 2 - 36 x^{2} + 25 + arcsin (\frac{6}{5} x)

= 2 - 36 x^{2} + 25 + arcsin (\frac{6}{5} x)

Add a constant to the solution

= 2 - 36 x^{2} + 25 + arcsin (\frac{6}{5} x) + C

What is the integral of (6-72x)/(sqrt(25-36x^2)) ?
The integral of (6-72x)/(sqrt(25-36x^2)) is 2sqrt(-36x^2+25)+arcsin(6/5 x)+C