What is the integral from 0 to 2 of x^3(7-x^2)^{1/2} ?

The integral from 0 to 2 of x^3(7-x^2)^{1/2} is (2(49*7^{1/2}-39*3^{1/2}))/(15)

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integral from 0 to 2 of x^3(7-x^2)^{1/2}

\int_{0}^{2} x^{3} (7 - x^{2})^{\frac{1}{2}} d x

\frac{2 ( 49 \cdot 7 ^{\frac{1}{2}} - 39 \cdot 3 ^{\frac{1}{2}} )}{15}

Decimal

8.27891 \dots

Solution steps

\int_{0}^{2} x^{3} (7 - x^{2})^{\frac{1}{2}} d x

Apply u-substitution

: \int_{7}^{3} - \frac{( - u + 7 ) u ^{\frac{1}{2}}}{2} d u

= \int_{7}^{3} - \frac{( - u + 7 ) u ^{\frac{1}{2}}}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{3}^{7} - \frac{( - u + 7 ) u ^{\frac{1}{2}}}{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_{3}^{7} (- u + 7) u^{\frac{1}{2}} d u)

Expand

(- u + 7) u^{\frac{1}{2}} : - u^{\frac{3}{2}} + 7 u^{\frac{1}{2}}

= - (- \frac{1}{2} \cdot \int_{3}^{7} - u^{\frac{3}{2}} + 7 u^{\frac{1}{2}} d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \frac{1}{2} (- \int_{3}^{7} u^{\frac{3}{2}} d u + \int_{3}^{7} 7 u^{\frac{1}{2}} d u))

\int_{3}^{7} u^{\frac{3}{2}} d u = \frac{98 \cdot 7 ^{\frac{1}{2}} - 18 \cdot 3 ^{\frac{1}{2}}}{5}

\int_{3}^{7} 7 u^{\frac{1}{2}} d u = 7 (\frac{14 \cdot 7 ^{\frac{1}{2}}}{3} - 2 \cdot 3^{\frac{1}{2}})

= - (- \frac{1}{2} (- \frac{98 \cdot 7 ^{\frac{1}{2}} - 18 \cdot 3 ^{\frac{1}{2}}}{5} + 7 (\frac{14 \cdot 7 ^{\frac{1}{2}}}{3} - 2 \cdot 3^{\frac{1}{2}})))

Simplify

- (- \frac{1}{2} (- \frac{98 \cdot 7 ^{\frac{1}{2}} - 18 \cdot 3 ^{\frac{1}{2}}}{5} + 7 (\frac{14 \cdot 7 ^{\frac{1}{2}}}{3} - 2 \cdot 3^{\frac{1}{2}}))) : \frac{2 ( 49 \cdot 7 ^{\frac{1}{2}} - 39 \cdot 3 ^{\frac{1}{2}} )}{15}

= \frac{2 ( 49 \cdot 7 ^{\frac{1}{2}} - 39 \cdot 3 ^{\frac{1}{2}} )}{15}

\int_{0}^{4} (x^{2} - 4 x + 3) d x

4 t \frac{d y}{d t} - 5 y = t

\int - 2 \cdot 3^{6 x} d x

y = cot^{2} (x)

\frac{\partial}{\partial x} (x^{2} + y^{2} - z)

What is the integral from 0 to 2 of x^3(7-x^2)^{1/2} ?
The integral from 0 to 2 of x^3(7-x^2)^{1/2} is (2(49*7^{1/2}-39*3^{1/2}))/(15)