What is the integral of (xsqrt(7-x^2)) ?

The integral of (xsqrt(7-x^2)) is -1/3 (7-x^2)^{3/2}+C

integral of (xsqrt(7-x^2))

\int (x 7 - x^{2}) d x

- \frac{1}{3} (7 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 7 - x^{2} d x

Apply u-substitution

= \int - \frac{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 u d u

Apply the Power Rule

= - \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 7 - x^{2}

= - \frac{1}{2} \cdot \frac{2}{3} (7 - x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{2} \cdot \frac{2}{3} (7 - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}} + C

x \to 4 lim (x^{3} - c x)

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

\frac{d}{d x} (e^{\frac{1}{x - 1}})

\int \frac{x x}{4 x ^{3}} d x

\int 1 + \frac{3}{2} x d x

What is the integral of (xsqrt(7-x^2)) ?
The integral of (xsqrt(7-x^2)) is -1/3 (7-x^2)^{3/2}+C