What is the integral of 4x^3arcsin(1/x) ?

The integral of 4x^3arcsin(1/x) is x^4arcsin(1/x)+1/3 (x^2-1)^{3/2}+sqrt(x^2-1)+C

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integral of 4x^3arcsin(1/x)

\int 4 x^{3} arcsin (\frac{1}{x}) d x

x^{4} arcsin (\frac{1}{x}) + \frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1 + C

Solution steps

\int 4 x^{3} arcsin (\frac{1}{x}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x^{3} arcsin (\frac{1}{x}) d x

Apply Integration By Parts

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

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

\int - \frac{x ^{2} x ^{2}}{4 x ^{2} - 1} d x = - \frac{1}{4} (\frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1)

= 4 (\frac{1}{4} x^{4} arcsin (\frac{1}{x}) - (- \frac{1}{4} (\frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1)))

Simplify

4 (\frac{1}{4} x^{4} arcsin (\frac{1}{x}) - (- \frac{1}{4} (\frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1))) : x^{4} arcsin (\frac{1}{x}) + \frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1

= x^{4} arcsin (\frac{1}{x}) + \frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1

Add a constant to the solution

= x^{4} arcsin (\frac{1}{x}) + \frac{1}{3} (x^{2} - 1)^{\frac{3}{2}} + x^{2} - 1 + C

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What is the integral of 4x^3arcsin(1/x) ?
The integral of 4x^3arcsin(1/x) is x^4arcsin(1/x)+1/3 (x^2-1)^{3/2}+sqrt(x^2-1)+C