What is the integral of sqrt(x)sqrt(100-x) ?

The integral of sqrt(x)sqrt(100-x) is 1250(arcsin(1/50 x-1)+1/2 sin(2arcsin(1/50 x-1)))+C

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integral of sqrt(x)sqrt(100-x)

\int x 100 - x d x

1250 (arcsin (\frac{1}{50} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} x - 1))) + C

Solution steps

\int x 100 - x d x

Apply exponent rule:

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

x 100 - x = x (100 - x)

= \int x (100 - x) d x

Expand

x (100 - x) : 100 x - x^{2}

= \int 100 x - x^{2} d x

Complete the square

100 x - x^{2} : - (x - 50)^{2} + 2500

= \int - (x - 50)^{2} + 2500 d x

Apply u-substitution

: \int - u^{2} + 2500 d u

= \int - u^{2} + 2500 d u

Apply Trigonometric Substitution

: \int 2500 cos^{2} (v) d v

= \int 2500 cos^{2} (v) d v

Take the constant out:

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

= 2500 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 2500 \cdot \int \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

= 2500 \cdot \frac{1}{2} \cdot \int 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

= 2500 \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 2500 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 2500 \cdot \frac{1}{2} (arcsin (\frac{1}{50} (x - 50)) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} (x - 50))))

Simplify

2500 \cdot \frac{1}{2} (arcsin (\frac{1}{50} (x - 50)) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} (x - 50)))) : 1250 (arcsin (\frac{1}{50} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} x - 1)))

= 1250 (arcsin (\frac{1}{50} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} x - 1)))

Add a constant to the solution

= 1250 (arcsin (\frac{1}{50} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{50} x - 1))) + C

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What is the integral of sqrt(x)sqrt(100-x) ?
The integral of sqrt(x)sqrt(100-x) is 1250(arcsin(1/50 x-1)+1/2 sin(2arcsin(1/50 x-1)))+C