What is the integral of tsin(kt) ?

The integral of tsin(kt) is -1/k tcos(kt)+1/(k^2)sin(kt)+C

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integral of tsin(kt)

\int t sin (k t) d t

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

Solution steps

\int t sin (k t) d t

Apply Integration By Parts

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

= - \frac{1}{k} t cos (k t) - \int - \frac{1}{k} cos (k t) d t

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{1}{y ^{2} y ^{2} + 16} d y

\int \frac{1}{x ^{2} - 5 ^{2}} d x

\int \frac{1}{2} x e^{- \frac{1}{2} x} d x

4 x^{\frac{5}{4}} + 10 x^{\frac{3}{2}} + 8 x

x \to 0 - lim ((1 + x)^{\frac{1}{x ^{2}}})

What is the integral of tsin(kt) ?
The integral of tsin(kt) is -1/k tcos(kt)+1/(k^2)sin(kt)+C