What is the integral of (x+1)/(sqrt(2-3x^2)) ?

The integral of (x+1)/(sqrt(2-3x^2)) is -1/3 sqrt(2-3x^2)+1/(sqrt(3))arcsin(sqrt(3/2)x)+C

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integral of (x+1)/(sqrt(2-3x^2))

\int \frac{x + 1}{2 - 3 x ^{2}} d x

- \frac{1}{3} 2 - 3 x^{2} + \frac{1}{3} arcsin (\frac{3}{2} x) + C

Solution steps

\int \frac{x + 1}{2 - 3 x ^{2}} d x

Expand

\frac{x + 1}{2 - 3 x ^{2}} : \frac{x}{2 - 3 x ^{2}} + \frac{1}{2 - 3 x ^{2}}

= \int \frac{x}{2 - 3 x ^{2}} + \frac{1}{2 - 3 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}{2 - 3 x ^{2}} d x + \int \frac{1}{2 - 3 x ^{2}} d x

\int \frac{x}{2 - 3 x ^{2}} d x = - \frac{1}{3} 2 - 3 x^{2}

\int \frac{1}{2 - 3 x ^{2}} d x = \frac{1}{3} arcsin (\frac{3}{2} x)

= - \frac{1}{3} 2 - 3 x^{2} + \frac{1}{3} arcsin (\frac{3}{2} x)

Add a constant to the solution

= - \frac{1}{3} 2 - 3 x^{2} + \frac{1}{3} arcsin (\frac{3}{2} x) + C

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What is the integral of (x+1)/(sqrt(2-3x^2)) ?
The integral of (x+1)/(sqrt(2-3x^2)) is -1/3 sqrt(2-3x^2)+1/(sqrt(3))arcsin(sqrt(3/2)x)+C