What is the integral from a to b of sqrt(3-x^2) ?

The integral from a to b of sqrt(3-x^2) is

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integral from a to b of sqrt(3-x^2)

\int_{a}^{b} 3 - x^{2} d x

\frac{3}{2} (arcsin (\frac{b}{3}) - arcsin (\frac{a}{3}) + \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3}))))

Solution steps

\int_{a}^{b} 3 - x^{2} d x

Apply Trigonometric Substitution

: \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} 3 cos^{2} (u) d u

= \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} 3 cos^{2} (u) d u

Take the constant out:

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

= 3 \cdot \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} cos^{2} (u) d u

Rewrite using trig identities

= 3 \cdot \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} 1 + cos (2 u) 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

= 3 \cdot \frac{1}{2} (\int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} 1 d u + \int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} cos (2 u) d u)

\int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} 1 d u = arcsin (\frac{b}{3}) - arcsin (\frac{a}{3})

\int_{a r c s i n (\frac{a}{3})}^{a r c s i n (\frac{b}{3})} cos (2 u) d u = \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3})))

= 3 \cdot \frac{1}{2} (arcsin (\frac{b}{3}) - arcsin (\frac{a}{3}) + \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3}))))

Simplify

3 \cdot \frac{1}{2} (arcsin (\frac{b}{3}) - arcsin (\frac{a}{3}) + \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3})))) : \frac{3}{2} (arcsin (\frac{b}{3}) - arcsin (\frac{a}{3}) + \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3}))))

= \frac{3}{2} (arcsin (\frac{b}{3}) - arcsin (\frac{a}{3}) + \frac{1}{2} (sin (2 arcsin (\frac{b}{3})) - sin (2 arcsin (\frac{a}{3}))))

\int_{- \frac{π}{2}}^{\frac{π}{2}} (cos (x))^{2} d x

\int_{3}^{4} 1

\int_{0}^{9} t d t

\int_{- \frac{π}{4}}^{\frac{π}{4}} 2 sec^{2} (x) d x

\int_{1}^{e^{2}} \frac{1}{x ( 1 + ln ( x ) )} d x

What is the integral from a to b of sqrt(3-x^2) ?
The integral from a to b of sqrt(3-x^2) is