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

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

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

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

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

Solution steps

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

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

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

\int 3 d x = 3 x

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

Add a constant to the solution

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

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