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

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

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integral of xsin(x^2+5)

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

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

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = x^{2} + 5

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{ln ( x )}{ln ( 7 )})

\int \frac{5}{6} x^{\frac{5}{6}} d x

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

x \to - 1 + lim (\frac{1}{x + 1})

x \to \infty + lim (e^{x} - 2 x)

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