What is the integral of 3tsin(t)cos(t) ?

The integral of 3tsin(t)cos(t) is 3/2 (-1/2 tcos(2t)+1/4 sin(2t))+C

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integral of 3tsin(t)cos(t)

\int 3 t sin (t) cos (t) d t

\frac{3}{2} (- \frac{1}{2} t cos (2 t) + \frac{1}{4} sin (2 t)) + C

Solution steps

\int 3 t sin (t) cos (t) d t

Take the constant out:

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

= 3 \cdot \int t sin (t) cos (t) d t

Rewrite using trig identities

= 3 \cdot \int \frac{1}{2} t sin (2 t) d t

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int t sin (2 t) d t

Apply Integration By Parts

: - \frac{1}{2} t cos (2 t) - \int - \frac{1}{2} cos (2 t) d t

= 3 \cdot \frac{1}{2} (- \frac{1}{2} t cos (2 t) - \int - \frac{1}{2} cos (2 t) d t)

\int - \frac{1}{2} cos (2 t) d t = - \frac{1}{4} sin (2 t)

= 3 \cdot \frac{1}{2} (- \frac{1}{2} t cos (2 t) - (- \frac{1}{4} sin (2 t)))

Simplify

3 \cdot \frac{1}{2} (- \frac{1}{2} t cos (2 t) - (- \frac{1}{4} sin (2 t))) : \frac{3}{2} (- \frac{1}{2} t cos (2 t) + \frac{1}{4} sin (2 t))

= \frac{3}{2} (- \frac{1}{2} t cos (2 t) + \frac{1}{4} sin (2 t))

Add a constant to the solution

= \frac{3}{2} (- \frac{1}{2} t cos (2 t) + \frac{1}{4} sin (2 t)) + C

(x^{4} + 2 x^{3} y^{2}) d x + x^{4} y d y = 0

\frac{d y}{d t} - 2 y = 3 e^{- 2 t}, y (0) = 10

\frac{\partial}{\partial x} (x^{3} + 2 x^{2} y + x + 2 y)

\int \frac{1}{X ^{2}} D X d X

n = 0 \sum \infty 20000

What is the integral of 3tsin(t)cos(t) ?
The integral of 3tsin(t)cos(t) is 3/2 (-1/2 tcos(2t)+1/4 sin(2t))+C