What is the integral of sqrt(75-10x-x^2) ?

The integral of sqrt(75-10x-x^2) is 50(arcsin(1/10 (x+5))+1/2 sin(2arcsin(1/10 (x+5))))+C

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integral of sqrt(75-10x-x^2)

\int 75 - 10 x - x^{2} d x

50 (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5)))) + C

Solution steps

\int 75 - 10 x - x^{2} d x

Complete the square

75 - 10 x - x^{2} : - (x + 5)^{2} + 100

= \int - (x + 5)^{2} + 100 d x

Apply u-substitution

: \int - u^{2} + 100 d u

= \int - u^{2} + 100 d u

Apply Trigonometric Substitution

: \int 100 cos^{2} (v) d v

= \int 100 cos^{2} (v) d v

Take the constant out:

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

= 100 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 100 \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

= 100 \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

= 100 \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)

= 100 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 100 \cdot \frac{1}{2} (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5))))

Simplify

100 \cdot \frac{1}{2} (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5)))) : 50 (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5))))

= 50 (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5))))

Add a constant to the solution

= 50 (arcsin (\frac{1}{10} (x + 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x + 5)))) + C

y^{^{''}} - y^{^{'}} - 2 y = - 4 t + 8 t^{2}

y = x - 2, y = x^{2} - 4, x = 5

\int y cos (z) d z

x y = (- 1 - x^{2}) y^{^{'}}

\int 2 x^{- \frac{1}{2}} d x

What is the integral of sqrt(75-10x-x^2) ?
The integral of sqrt(75-10x-x^2) is 50(arcsin(1/10 (x+5))+1/2 sin(2arcsin(1/10 (x+5))))+C