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

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

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integral of xsin((x^2)/2)

\int x sin (\frac{x ^{2}}{2}) d x

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

Solution steps

\int x sin (\frac{x ^{2}}{2}) d x

Apply u-substitution

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

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

Apply u-substitution

: \int sin (v) \cdot 2 d v

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

Take the constant out:

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

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

Use the common integral:

\int sin (v) d v = - cos (v)

= \frac{1}{2} \cdot 2 (- cos (v))

Substitute back

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

Simplify

\frac{1}{2} \cdot 2 (- cos (\frac{x ^{2}}{2})) : - cos (\frac{x ^{2}}{2})

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

Add a constant to the solution

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

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