What is the integral from 0 to 5 of sqrt(x)sqrt(5-x) ?

The integral from 0 to 5 of sqrt(x)sqrt(5-x) is (25pi)/8

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integral from 0 to 5 of sqrt(x)sqrt(5-x)

\int_{0}^{5} x 5 - x d x

\frac{25 π}{8}

Decimal Notation

9.81747 \dots

Solution steps

\int_{0}^{5} x 5 - x d x

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x 5 - x = x (5 - x)

= \int_{0}^{5} x (5 - x) d x

Expand

x (5 - x) : 5 x - x^{2}

= \int_{0}^{5} 5 x - x^{2} d x

Complete the square

5 x - x^{2} : - (x - \frac{5}{2})^{2} + \frac{25}{4}

= \int_{0}^{5} - (x - \frac{5}{2})^{2} + \frac{25}{4} d x

Apply u-substitution

: \int_{- \frac{5}{2}}^{\frac{5}{2}} \frac{- 4 u ^{2} + 25}{2} d u

= \int_{- \frac{5}{2}}^{\frac{5}{2}} \frac{- 4 u ^{2} + 25}{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_{- \frac{5}{2}}^{\frac{5}{2}} - 4 u^{2} + 25 d u

Apply Trigonometric Substitution

: \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{25}{2} cos^{2} (v) d v

= \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{25}{2} cos^{2} (v) d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{25}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (v) d v

Rewrite using trig identities

= \frac{1}{2} \cdot \frac{25}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{25}{2} \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 v) d v

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} \cdot \frac{25}{2} \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v = 0

= \frac{1}{2} \cdot \frac{25}{2} \cdot \frac{1}{2} (π + 0)

Simplify

\frac{1}{2} \cdot \frac{25}{2} \cdot \frac{1}{2} (π + 0) : \frac{25 π}{8}

= \frac{25 π}{8}

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What is the integral from 0 to 5 of sqrt(x)sqrt(5-x) ?
The integral from 0 to 5 of sqrt(x)sqrt(5-x) is (25pi)/8