What is the integral from 0 to 1 of 100xsin(3pix) ?

The integral from 0 to 1 of 100xsin(3pix) is (100)/(3pi)

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integral from 0 to 1 of 100xsin(3pix)

\int_{0}^{1} 100 x sin (3 π x) d x

\frac{100}{3 π}

Decimal

10.61032 \dots

Solution steps

\int_{0}^{1} 100 x sin (3 π x) d x

Take the constant out:

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

= 100 \cdot \int_{0}^{1} x sin (3 π x) d x

Apply Integration By Parts

: [- \frac{1}{3 π} x cos (3 π x) - \int - \frac{1}{3 π} cos (3 π x) d x]_{0}^{1}

= 100 [- \frac{1}{3 π} x cos (3 π x) - \int - \frac{1}{3 π} cos (3 π x) d x]_{0}^{1}

\int - \frac{1}{3 π} cos (3 π x) d x = - \frac{1}{9 π ^{2}} sin (3 π x)

= 100 [- \frac{1}{3 π} x cos (3 π x) - (- \frac{1}{9 π ^{2}} sin (3 π x))]_{0}^{1}

Simplify

= 100 [- \frac{1}{3 π} x cos (3 π x) + \frac{1}{9 π ^{2}} sin (3 π x)]_{0}^{1}

Compute the boundaries:

\frac{1}{3 π}

= 100 \cdot \frac{1}{3 π}

Simplify

= \frac{100}{3 π}

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What is the integral from 0 to 1 of 100xsin(3pix) ?
The integral from 0 to 1 of 100xsin(3pix) is (100)/(3pi)