What is the integral of xsin(2x^2-5) ?

The integral of xsin(2x^2-5) is -1/4 cos(2x^2-5)+C

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integral of xsin(2x^2-5)

\int x sin (2 x^{2} - 5) d x

- \frac{1}{4} cos (2 x^{2} - 5) + C

Solution steps

\int x sin (2 x^{2} - 5) d x

Apply u-substitution

: \int \frac{sin ( 2 u - 5 )}{2} d u

= \int \frac{sin ( 2 u - 5 )}{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 (2 u - 5) d u

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

= \frac{1}{2} \cdot \frac{1}{2} (- cos (2 x^{2} - 5))

Simplify

\frac{1}{2} \cdot \frac{1}{2} (- cos (2 x^{2} - 5)) : - \frac{1}{4} cos (2 x^{2} - 5)

= - \frac{1}{4} cos (2 x^{2} - 5)

Add a constant to the solution

= - \frac{1}{4} cos (2 x^{2} - 5) + C

\int sin^{2} (x) + sec^{2} (5 x) d x

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\int ln^{2} (\frac{4}{x}) d x

What is the integral of xsin(2x^2-5) ?
The integral of xsin(2x^2-5) is -1/4 cos(2x^2-5)+C