What is the integral of sqrt(3/4-x^2) ?

The integral of sqrt(3/4-x^2) is 3/8 (arcsin(2/(sqrt(3))x)+1/2 sin(2arcsin(2/(sqrt(3))x)))+C

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integral of sqrt(3/4-x^2)

\int \frac{3}{4} - x^{2} d x

\frac{3}{8} (arcsin (\frac{2}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{3} x))) + C

Solution steps

\int \frac{3}{4} - x^{2} d x

Apply Trigonometric Substitution

: \int \frac{3 cos ^{2} ( u )}{4} d u

= \int \frac{3 cos ^{2} ( u )}{4} d u

Take the constant out:

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

= \frac{3}{4} \cdot \int cos^{2} (u) d u

Rewrite using trig identities

= \frac{3}{4} \cdot \int \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

= \frac{3}{4} \cdot \frac{1}{2} \cdot \int 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

= \frac{3}{4} \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

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

= \frac{3}{4} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{\frac{3}{2}} x)

= \frac{3}{4} \cdot \frac{1}{2} (arcsin (\frac{1}{\frac{3}{2}} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{\frac{3}{2}} x)))

Simplify

\frac{3}{4} \cdot \frac{1}{2} (arcsin (\frac{1}{\frac{3}{2}} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{\frac{3}{2}} x))) : \frac{3}{8} (arcsin (\frac{2}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{3} x)))

= \frac{3}{8} (arcsin (\frac{2}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{3} x)))

Add a constant to the solution

= \frac{3}{8} (arcsin (\frac{2}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{3} x))) + C

\frac{d}{d x} (x^{2} + 7 x)

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

\int \frac{x ^{7} e ^{x} - 8 x ^{6}}{x ^{7}} d x

\frac{d}{d x} (lo g_{18} (x^{2} - 5 x))

\frac{x}{e ^{x} - 1}

What is the integral of sqrt(3/4-x^2) ?
The integral of sqrt(3/4-x^2) is 3/8 (arcsin(2/(sqrt(3))x)+1/2 sin(2arcsin(2/(sqrt(3))x)))+C