What is the integral of sqrt(x^7)sin(1+x^{9/2}) ?

The integral of sqrt(x^7)sin(1+x^{9/2}) is -2/9 cos(1+x^{9/2})+C

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integral of sqrt(x^7)sin(1+x^{9/2})

\int x^{7} sin (1 + x^{\frac{9}{2}}) d x

- \frac{2}{9} cos (1 + x^{\frac{9}{2}}) + C

Solution steps

\int x^{7} sin (1 + x^{\frac{9}{2}}) d x

Apply u-substitution

= \int \frac{2 sin ( u )}{9} d u

Take the constant out:

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

= \frac{2}{9} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2}{9} (- cos (u))

Substitute back

u = 1 + x^{\frac{9}{2}}

= \frac{2}{9} (- cos (1 + x^{\frac{9}{2}}))

Simplify

= - \frac{2}{9} cos (1 + x^{\frac{9}{2}})

Add a constant to the solution

= - \frac{2}{9} cos (1 + x^{\frac{9}{2}}) + C

What is the integral of sqrt(x^7)sin(1+x^{9/2}) ?
The integral of sqrt(x^7)sin(1+x^{9/2}) is -2/9 cos(1+x^{9/2})+C