What is the integral of sqrt(1+3cos^2(x))sin(2x) ?

The integral of sqrt(1+3cos^2(x))sin(2x) is -2/9 (1+3cos^2(x))^{3/2}+C

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integral of sqrt(1+3cos^2(x))sin(2x)

\int 1 + 3 cos^{2} (x) sin (2 x) d x

- \frac{2}{9} (1 + 3 cos^{2} (x))^{\frac{3}{2}} + C

Solution steps

\int 1 + 3 cos^{2} (x) sin (2 x) d x

Apply u-substitution

: \int - \frac{u}{3} d u

= \int - \frac{u}{3} 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 u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + 3 cos^{2} (x)

= - \frac{1}{3} \cdot \frac{2}{3} (1 + 3 cos^{2} (x))^{\frac{3}{2}}

Simplify

- \frac{1}{3} \cdot \frac{2}{3} (1 + 3 cos^{2} (x))^{\frac{3}{2}} : - \frac{2}{9} (1 + 3 cos^{2} (x))^{\frac{3}{2}}

= - \frac{2}{9} (1 + 3 cos^{2} (x))^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{9} (1 + 3 cos^{2} (x))^{\frac{3}{2}} + C

f (x) = 12 - 4 e^{x}, (0, 8)

\frac{d}{d z} (\frac{x ln ( \frac{x y}{z} )}{z})

\frac{d}{d x} (3 x + 7 x)

\frac{z}{z + \frac{a}{z}}

\int_{0}^{x^{2} + 3} \frac{1}{x ^{2} + 3} d x

What is the integral of sqrt(1+3cos^2(x))sin(2x) ?
The integral of sqrt(1+3cos^2(x))sin(2x) is -2/9 (1+3cos^2(x))^{3/2}+C