What is the integral of sqrt(1-3x^2) ?

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

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integral of sqrt(1-3x^2)

\int 1 - 3 x^{2} d x

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

Solution steps

\int 1 - 3 x^{2} d x

Apply Trigonometric Substitution

: \int \frac{1}{3} cos^{2} (u) d u

= \int \frac{1}{3} cos^{2} (u) d u

Take the constant out:

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

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

Rewrite using trig identities

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

Substitute back

u = arcsin (3 x)

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

Simplify

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

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

Add a constant to the solution

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

\int tan^{2} (6 x) d x

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\frac{m P}{P + ( m - P ) e ^{- km t}}

\frac{d ^{2}}{d x ^{2}} (\frac{3}{4 t ^{5}})

\int \frac{4}{x ^{4} - 1} d x

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