What is the integral of xsin(bx) ?

The integral of xsin(bx) is -1/b xcos(bx)+1/(b^2)sin(bx)+C

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integral of xsin(bx)

\int x sin (b x) d x

- \frac{1}{b} x cos (b x) + \frac{1}{b ^{2}} sin (b x) + C

Solution steps

\int x sin (b x) d x

Apply Integration By Parts

: - \frac{1}{b} x cos (b x) - \int - \frac{1}{b} cos (b x) d x

= - \frac{1}{b} x cos (b x) - \int - \frac{1}{b} cos (b x) d x

\int - \frac{1}{b} cos (b x) d x = - \frac{1}{b ^{2}} sin (b x)

= - \frac{1}{b} x cos (b x) - (- \frac{1}{b ^{2}} sin (b x))

Simplify

= - \frac{1}{b} x cos (b x) + \frac{1}{b ^{2}} sin (b x)

Add a constant to the solution

= - \frac{1}{b} x cos (b x) + \frac{1}{b ^{2}} sin (b x) + C

\int - x^{2} + 3 d x

\frac{d}{d x} (x x - 3)

\int 3 sin (2 t) d t

\int ln (x^{2} + 5) d x

f (x) = 4 x^{2} - 4 x + 1, at x = 1

What is the integral of xsin(bx) ?
The integral of xsin(bx) is -1/b xcos(bx)+1/(b^2)sin(bx)+C