What is the integral of xsin(sqrt(x^2+4)) ?

The integral of xsin(sqrt(x^2+4)) is -cos(sqrt(x^2+4))sqrt(x^2+4)+sin(sqrt(x^2+4))+C

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integral of xsin(sqrt(x^2+4))

\int x sin (x^{2} + 4) d x

- cos (x^{2} + 4) x^{2} + 4 + sin (x^{2} + 4) + C

Solution steps

\int x sin (x^{2} + 4) d x

Apply u-substitution

: \int \frac{sin ( u + 4 )}{2} d u

= \int \frac{sin ( u + 4 )}{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 + 4) d u

Apply u-substitution

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

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

Apply Integration By Parts

: - v cos (v) - \int - cos (v) d v

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

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

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

Substitute back

= \frac{1}{2} \cdot 2 (- x^{2} + 4 cos (x^{2} + 4) - (- sin (x^{2} + 4)))

Simplify

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

= - cos (x^{2} + 4) x^{2} + 4 + sin (x^{2} + 4)

Add a constant to the solution

= - cos (x^{2} + 4) x^{2} + 4 + sin (x^{2} + 4) + C

(x + 1)^{81}

\frac{e ^{2 x}}{ln ( 6 x )}

n = 3 \sum \infty \frac{1}{n ( n - 1 )}

\frac{d}{d x} (sin^{3} (3 x + 2))

\frac{- x + ln ( 1 + x )}{x ^{2}}

What is the integral of xsin(sqrt(x^2+4)) ?
The integral of xsin(sqrt(x^2+4)) is -cos(sqrt(x^2+4))sqrt(x^2+4)+sin(sqrt(x^2+4))+C