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

The integral from 0 to 1 of xcos(3pix) is -2/(9pi^2)

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integral from 0 to 1 of xcos(3pix)

\int_{0}^{1} x cos (3 π x) d x

- \frac{2}{9 π ^{2}}

Decimal Notation

- 0.02251 \dots

Solution steps

\int_{0}^{1} x cos (3 π x) d x

Apply u-substitution

: \int_{0}^{3 π} \frac{u cos ( u )}{9 π ^{2}} d u

= \int_{0}^{3 π} \frac{u cos ( u )}{9 π ^{2}} d u

Take the constant out:

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

= \frac{1}{9 π ^{2}} \cdot \int_{0}^{3 π} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{3 π}

= \frac{1}{9 π ^{2}} [u sin (u) - \int sin (u) d u]_{0}^{3 π}

\int sin (u) d u = - cos (u)

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

Simplify

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

Compute the boundaries:

- 2

= \frac{1}{9 π ^{2}} (- 2)

Simplify

= - \frac{2}{9 π ^{2}}

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What is the integral from 0 to 1 of xcos(3pix) ?
The integral from 0 to 1 of xcos(3pix) is -2/(9pi^2)