What is the integral of xsin(x^2-1) ?

The integral of xsin(x^2-1) is 1/2 (x^2sin(x^2-1)+x^2sin(1-x^2)-cos(1-x^2))+C

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integral of xsin(x^2-1)

\int x sin (x^{2} - 1) d x

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

Solution steps

\int x sin (x^{2} - 1) d x

Apply u-substitution

: \int \frac{sin ( u - 1 )}{2} d u

= \int \frac{sin ( u - 1 )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (u - 1) d u

Apply Integration By Parts

: sin (u - 1) u - \int cos (u - 1) u d u

= \frac{1}{2} (sin (u - 1) u - \int cos (u - 1) u d u)

\int cos (u - 1) u d u = - u sin (1 - u) + cos (1 - u)

= \frac{1}{2} (sin (u - 1) u - (- u sin (1 - u) + cos (1 - u)))

Substitute back

u = x^{2}

= \frac{1}{2} (sin (x^{2} - 1) x^{2} - (- x^{2} sin (1 - x^{2}) + cos (1 - x^{2})))

Expand

sin (x^{2} - 1) x^{2} - (- x^{2} sin (1 - x^{2}) + cos (1 - x^{2})) : sin (x^{2} - 1) x^{2} + x^{2} sin (1 - x^{2}) - cos (1 - x^{2})

= \frac{1}{2} (x^{2} sin (x^{2} - 1) + x^{2} sin (1 - x^{2}) - cos (1 - x^{2}))

Add a constant to the solution

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

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What is the integral of xsin(x^2-1) ?
The integral of xsin(x^2-1) is 1/2 (x^2sin(x^2-1)+x^2sin(1-x^2)-cos(1-x^2))+C