What is the integral of 7/5 sin(3x)cos(x/3) ?

The integral of 7/5 sin(3x)cos(x/3) is 21/10 (-1/10 cos((10x)/3)-1/8 cos((8x)/3))+C

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integral of 7/5 sin(3x)cos(x/3)

\int \frac{7}{5} sin (3 x) cos (\frac{x}{3}) d x

\frac{21}{10} (- \frac{1}{10} cos (\frac{10 x}{3}) - \frac{1}{8} cos (\frac{8 x}{3})) + C

Solution steps

\int \frac{7}{5} sin (3 x) cos (\frac{x}{3}) d x

Take the constant out:

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

= \frac{7}{5} \cdot \int sin (3 x) cos (\frac{x}{3}) d x

Apply u-substitution

: \int 3 sin (9 u) cos (u) d u

= \frac{7}{5} \cdot \int 3 sin (9 u) cos (u) d u

Take the constant out:

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

= \frac{7}{5} \cdot 3 \cdot \int sin (9 u) cos (u) d u

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \frac{7}{5} \cdot 3 \cdot \int \frac{sin ( 9 u + u ) + sin ( 9 u - u )}{2} d u

Take the constant out:

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

= \frac{7}{5} \cdot 3 \cdot \frac{1}{2} \cdot \int sin (9 u + u) + sin (9 u - 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{7}{5} \cdot 3 \cdot \frac{1}{2} (\int sin (9 u + u) d u + \int sin (9 u - u) d u)

\int sin (9 u + u) d u = - \frac{1}{10} cos (10 u)

\int sin (9 u - u) d u = - \frac{1}{8} cos (8 u)

= \frac{7}{5} \cdot 3 \cdot \frac{1}{2} (- \frac{1}{10} cos (10 u) - \frac{1}{8} cos (8 u))

Substitute back

u = \frac{x}{3}

= \frac{7}{5} \cdot 3 \cdot \frac{1}{2} (- \frac{1}{10} cos (10 \cdot \frac{x}{3}) - \frac{1}{8} cos (8 \cdot \frac{x}{3}))

Simplify

\frac{7}{5} \cdot 3 \cdot \frac{1}{2} (- \frac{1}{10} cos (10 \cdot \frac{x}{3}) - \frac{1}{8} cos (8 \cdot \frac{x}{3})) : \frac{21}{10} (- \frac{1}{10} cos (\frac{10 x}{3}) - \frac{1}{8} cos (\frac{8 x}{3}))

= \frac{21}{10} (- \frac{1}{10} cos (\frac{10 x}{3}) - \frac{1}{8} cos (\frac{8 x}{3}))

Add a constant to the solution

= \frac{21}{10} (- \frac{1}{10} cos (\frac{10 x}{3}) - \frac{1}{8} cos (\frac{8 x}{3})) + C

\frac{\partial}{\partial x} (\frac{2 y}{x})

3 x x

\int_{- 7}^{7} 1 + x^{2} d x

x \to 0 lim (\frac{4}{3 + e ^{\frac{1}{x}}})

\frac{d}{d x} (sin (2 x + 3 y))

What is the integral of 7/5 sin(3x)cos(x/3) ?
The integral of 7/5 sin(3x)cos(x/3) is 21/10 (-1/10 cos((10x)/3)-1/8 cos((8x)/3))+C