What is the integral of sin(2x)cos(x) ?

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

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integral of sin(2x)cos(x)

\int sin (2 x) cos (x) d x

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

Solution steps

\int sin (2 x) cos (x) d x

Rewrite using trig identities

= \int 2 cos^{2} (x) sin (x) d x

Take the constant out:

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

= 2 \cdot \int cos^{2} (x) sin (x) d x

Apply u-substitution

: \int - u^{2} d u

= 2 \cdot \int - u^{2} d u

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = cos (x)

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

Simplify

2 (- \frac{cos ^{3} ( x )}{3}) : - \frac{2}{3} cos^{3} (x)

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

Add a constant to the solution

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

f (x) = x^{2} + 8

y = 2 x^{2} e^{x} + 4 x^{3}

x \to 0 lim ((1 - 5 x)^{\frac{7}{x}})

\frac{x}{x + 13}

x \to - 5 lim (\frac{x ^{2} + 6}{x + 2})

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