What is the integral from 0 to pi/2 of xsin(4x) ?

The integral from 0 to pi/2 of xsin(4x) is -pi/8

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integral from 0 to pi/2 of xsin(4x)

\int_{0}^{\frac{π}{2}} x sin (4 x) d x

- \frac{π}{8}

Decimal

- 0.39269 \dots

Solution steps

\int_{0}^{\frac{π}{2}} x sin (4 x) d x

Apply Integration By Parts

: [\frac{1}{4} (- x cos (4 x) - 4 \cdot \int - \frac{1}{4} cos (4 x) d x)]_{0}^{\frac{π}{2}}

= [\frac{1}{4} (- x cos (4 x) - 4 \cdot \int - \frac{1}{4} cos (4 x) d x)]_{0}^{\frac{π}{2}}

\int - \frac{1}{4} cos (4 x) d x = - \frac{1}{16} sin (4 x)

= [\frac{1}{4} (- x cos (4 x) - 4 (- \frac{1}{16} sin (4 x)))]_{0}^{\frac{π}{2}}

Simplify

[\frac{1}{4} (- x cos (4 x) - 4 (- \frac{1}{16} sin (4 x)))]_{0}^{\frac{π}{2}} : [\frac{1}{4} (- x cos (4 x) + \frac{1}{4} sin (4 x))]_{0}^{\frac{π}{2}}

= [\frac{1}{4} (- x cos (4 x) + \frac{1}{4} sin (4 x))]_{0}^{\frac{π}{2}}

Compute the boundaries:

- \frac{π}{8}

= - \frac{π}{8}

x \to 5 - lim (\frac{5}{( x - 5 ) ^{3}})

4 x

\int 3 + s (s + 1)^{2} d s

f (x) = ln (25 sin^{2} (x))

y (X) = 0.167 X

What is the integral from 0 to pi/2 of xsin(4x) ?
The integral from 0 to pi/2 of xsin(4x) is -pi/8