What is the integral from 0 to 2 of cos(x^2)x ?

The integral from 0 to 2 of cos(x^2)x is 1/2 sin(4)

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integral from 0 to 2 of cos(x^2)x

\int_{0}^{2} cos (x^{2}) x d x

\frac{1}{2} sin (4)

Decimal

- 0.37840 \dots

Solution steps

\int_{0}^{2} cos (x^{2}) x d x

Apply u-substitution

: \int_{0}^{4} \frac{cos ( u )}{2} d u

= \int_{0}^{4} \frac{cos ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{4} cos (u) d u

Use the common integral:

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

= \frac{1}{2} [sin (u)]_{0}^{4}

Compute the boundaries:

sin (4)

= \frac{1}{2} sin (4)

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What is the integral from 0 to 2 of cos(x^2)x ?
The integral from 0 to 2 of cos(x^2)x is 1/2 sin(4)