What is the integral of (9-162x)/(sqrt(1-81x^2)) ?

The integral of (9-162x)/(sqrt(1-81x^2)) is arcsin(9x)+2sqrt(1-81x^2)+C

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integral of (9-162x)/(sqrt(1-81x^2))

\int \frac{9 - 162 x}{1 - 81 x ^{2}} d x

arcsin (9 x) + 2 1 - 81 x^{2} + C

Solution steps

\int \frac{9 - 162 x}{1 - 81 x ^{2}} d x

Expand

\frac{9 - 162 x}{1 - 81 x ^{2}} : \frac{9}{1 - 81 x ^{2}} - \frac{162 x}{1 - 81 x ^{2}}

= \int \frac{9}{1 - 81 x ^{2}} - \frac{162 x}{1 - 81 x ^{2}} d x

Apply the Sum Rule:

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

= \int \frac{9}{1 - 81 x ^{2}} d x - \int \frac{162 x}{1 - 81 x ^{2}} d x

\int \frac{9}{1 - 81 x ^{2}} d x = arcsin (9 x)

\int \frac{162 x}{1 - 81 x ^{2}} d x = - 2 1 - 81 x^{2}

= arcsin (9 x) - (- 2 1 - 81 x^{2})

Simplify

= arcsin (9 x) + 2 1 - 81 x^{2}

Add a constant to the solution

= arcsin (9 x) + 2 1 - 81 x^{2} + C

y = a^{5} + cos^{5} (x)

\frac{d}{d x} (\frac{2 x ^{\frac{5}{2}}}{5})

y = x^{2} - 2 x - 3, (2, - 3)

\frac{\partial}{\partial x} (x y)

f (x) = x^{4} - 4 x^{3} + 8, at x = 2

What is the integral of (9-162x)/(sqrt(1-81x^2)) ?
The integral of (9-162x)/(sqrt(1-81x^2)) is arcsin(9x)+2sqrt(1-81x^2)+C