What is the integral of ((x-1))/(sqrt(81-4x^2)) ?

The integral of ((x-1))/(sqrt(81-4x^2)) is -1/4 sqrt(81-4x^2)-1/2 arcsin(2/9 x)+C

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integral of ((x-1))/(sqrt(81-4x^2))

\int \frac{( x - 1 )}{81 - 4 x ^{2}} d x

- \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x) + C

Solution steps

\int \frac{x - 1}{81 - 4 x ^{2}} d x

Expand

\frac{x - 1}{81 - 4 x ^{2}} : \frac{x}{81 - 4 x ^{2}} - \frac{1}{81 - 4 x ^{2}}

= \int \frac{x}{81 - 4 x ^{2}} - \frac{1}{81 - 4 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{x}{81 - 4 x ^{2}} d x - \int \frac{1}{81 - 4 x ^{2}} d x

\int \frac{x}{81 - 4 x ^{2}} d x = - \frac{1}{4} 81 - 4 x^{2}

\int \frac{1}{81 - 4 x ^{2}} d x = \frac{1}{2} arcsin (\frac{2}{9} x)

= - \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x)

Add a constant to the solution

= - \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x) + C

What is the integral of ((x-1))/(sqrt(81-4x^2)) ?
The integral of ((x-1))/(sqrt(81-4x^2)) is -1/4 sqrt(81-4x^2)-1/2 arcsin(2/9 x)+C