What is the integral of sin(x)cos(x)sqrt(1-sin^4(x)) ?

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

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integral of sin(x)cos(x)sqrt(1-sin^4(x))

\int sin (x) cos (x) 1 - sin^{4} (x) d x

\frac{1}{4} (arcsin (sin^{2} (x)) + \frac{1}{2} sin (2 arcsin (sin^{2} (x)))) + C

Solution steps

\int sin (x) cos (x) 1 - sin^{4} (x) d x

Apply u-substitution

: \int u 1 - u^{4} d u

= \int u 1 - u^{4} d u

Simplify

u^{4} : (u^{2})^{2}

= \int u 1 - (u^{2})^{2} d u

Apply u-substitution

: \int \frac{1 - v ^{2}}{2} d v

= \int \frac{1 - v ^{2}}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 - v^{2} d v

Apply Trigonometric Substitution

: \int cos^{2} (w) d w

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

Rewrite using trig identities

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

Take the constant out:

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

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

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}{2} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{4} (arcsin (sin^{2} (x)) + \frac{1}{2} sin (2 arcsin (sin^{2} (x)))) + C

\frac{\partial}{\partial x} ((3 y) \cdot (2 x)^{- 1})

f (x) = sin (tan (2 x))

2 (2 x^{5} - 3 x + 2)^{8}

\int 3 x y

\int \frac{1}{( 2 - x ) 1 - x} d x

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