What is the integral of x*arcsin(x) ?

The integral of x*arcsin(x) is 1/2 x^2arcsin(x)-1/4 (arcsin(x)-1/2 sin(2arcsin(x)))+C

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integral of x*arcsin(x)

\int x \cdot arcsin (x) d x

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

Solution steps

\int x arcsin (x) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

(x^{2} + x^{3})^{4}

\int \frac{t}{( t ^{2} + 1 ) ^{\frac{7}{2}}} d t

y = 2 x^{2} - 1, (0, 3)

y^{^{''}} + 4 y^{^{'}} + 4 y = 4 e^{- 2 t}

\int \frac{( x - 3 )}{( x + 4 ) ( x - 5 )} d x

What is the integral of x*arcsin(x) ?
The integral of x*arcsin(x) is 1/2 x^2arcsin(x)-1/4 (arcsin(x)-1/2 sin(2arcsin(x)))+C