What is the integral of sqrt(x^5)sin(3+x^{7/2}) ?

The integral of sqrt(x^5)sin(3+x^{7/2}) is -2/7 cos(3+x^{7/2})+C

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integral of sqrt(x^5)sin(3+x^{7/2})

\int x^{5} sin (3 + x^{\frac{7}{2}}) d x

- \frac{2}{7} cos (3 + x^{\frac{7}{2}}) + C

Solution steps

\int x^{5} sin (3 + x^{\frac{7}{2}}) d x

x^{5} = x^{\frac{5}{2}},

assuming

x \geq 0

= \int x^{\frac{5}{2}} sin (3 + x^{\frac{7}{2}}) d x

Apply u-substitution

: \int \frac{2 sin ( u )}{7} d u

= \int \frac{2 sin ( u )}{7} d u

Take the constant out:

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

= \frac{2}{7} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2}{7} (- cos (u))

Substitute back

u = 3 + x^{\frac{7}{2}}

= \frac{2}{7} (- cos (3 + x^{\frac{7}{2}}))

Simplify

= - \frac{2}{7} cos (3 + x^{\frac{7}{2}})

Add a constant to the solution

= - \frac{2}{7} cos (3 + x^{\frac{7}{2}}) + C

\frac{\partial}{\partial y} (ln (\frac{x + y}{x - y}))

f (x) = \frac{7}{x}, at x = 2 or, x = 7

\int \frac{2 cos ( x )}{sin ( x )} d x

\frac{d}{d x} (x^{2} + 4 y, 3 x)

\int_{0}^{\frac{π}{2}} cos^{3} (15 x) d x

What is the integral of sqrt(x^5)sin(3+x^{7/2}) ?
The integral of sqrt(x^5)sin(3+x^{7/2}) is -2/7 cos(3+x^{7/2})+C