What is the integral of 11sin^{-3/2}(x)cos^3(x) ?

The integral of 11sin^{-3/2}(x)cos^3(x) is 11(-2/(sqrt(sin(x)))-2/3 sin^{3/2}(x))+C

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integral of 11sin^{-3/2}(x)cos^3(x)

\int 11 sin^{- \frac{3}{2}} (x) cos^{3} (x) d x

11 (- \frac{2}{sin ( x )} - \frac{2}{3} sin^{\frac{3}{2}} (x)) + C

Solution steps

\int 11 sin^{- \frac{3}{2}} (x) cos^{3} (x) d x

Take the constant out:

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

= 11 \cdot \int sin^{- \frac{3}{2}} (x) cos^{3} (x) d x

Rewrite using trig identities

= 11 \cdot \int (1 - sin^{2} (x)) cos (x) sin^{- \frac{3}{2}} (x) d x

Apply u-substitution

: \int \frac{1 - u ^{2}}{u ^{\frac{3}{2}}} d u

= 11 \cdot \int \frac{1 - u ^{2}}{u ^{\frac{3}{2}}} d u

Expand

\frac{1 - u ^{2}}{u ^{\frac{3}{2}}} : \frac{1}{u ^{\frac{3}{2}}} - u^{\frac{1}{2}}

= 11 \cdot \int \frac{1}{u ^{\frac{3}{2}}} - u^{\frac{1}{2}} 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

= 11 (\int \frac{1}{u ^{\frac{3}{2}}} d u - \int u^{\frac{1}{2}} d u)

\int \frac{1}{u ^{\frac{3}{2}}} d u = - \frac{2}{u}

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

= 11 (- \frac{2}{u} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = sin (x)

= 11 (- \frac{2}{sin ( x )} - \frac{2}{3} sin^{\frac{3}{2}} (x))

Add a constant to the solution

= 11 (- \frac{2}{sin ( x )} - \frac{2}{3} sin^{\frac{3}{2}} (x)) + C

\int \frac{2 ^{x} 3 ^{x}}{9 ^{x} - 4 ^{x}} d x

x \to - 1 lim (x^{2} - 5 x + 7)

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

y^{^{'}} = x y^{- 2} - 2 y

x^{2} y^{^{'}} + x y = \frac{y ^{3}}{x}

What is the integral of 11sin^{-3/2}(x)cos^3(x) ?
The integral of 11sin^{-3/2}(x)cos^3(x) is 11(-2/(sqrt(sin(x)))-2/3 sin^{3/2}(x))+C