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

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

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integral of tsin(t)cos(t)

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

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

Solution steps

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

Rewrite using trig identities

= \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

= \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

= \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)

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

Simplify

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

Add a constant to the solution

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

x \to \infty lim (x y)

\frac{d y}{d x} + \frac{y}{x} = 4 x^{8} y^{2}

\frac{\partial}{\partial x} (- x^{4})

x^{2} y^{^{''}} + 3 x y^{^{'}} + y = 0, y (1) = 3.6, y^{^{'}} (1) = 0.4

\int_{- 1}^{7} 3 x + 7 d x

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