What is the integral of sqrt(x^{19)}sin(5+x^{21/2}) ?

The integral of sqrt(x^{19)}sin(5+x^{21/2}) is -2/21 cos(5+x^{21/2})+C

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integral of sqrt(x^{19)}sin(5+x^{21/2})

\int x^{19} sin (5 + x^{\frac{21}{2}}) d x

- \frac{2}{21} cos (5 + x^{\frac{21}{2}}) + C

Solution steps

\int x^{19} sin (5 + x^{\frac{21}{2}}) d x

x^{19} = x^{\frac{19}{2}},

assuming

x \geq 0

= \int x^{\frac{19}{2}} sin (5 + x^{\frac{21}{2}}) d x

Apply u-substitution

: \int \frac{2 sin ( u )}{21} d u

= \int \frac{2 sin ( u )}{21} d u

Take the constant out:

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

= \frac{2}{21} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2}{21} (- cos (u))

Substitute back

u = 5 + x^{\frac{21}{2}}

= \frac{2}{21} (- cos (5 + x^{\frac{21}{2}}))

Simplify

= - \frac{2}{21} cos (5 + x^{\frac{21}{2}})

Add a constant to the solution

= - \frac{2}{21} cos (5 + x^{\frac{21}{2}}) + C

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What is the integral of sqrt(x^{19)}sin(5+x^{21/2}) ?
The integral of sqrt(x^{19)}sin(5+x^{21/2}) is -2/21 cos(5+x^{21/2})+C