What is the integral from 0 to 2 of 4xcos(x^2) ?

The integral from 0 to 2 of 4xcos(x^2) is 2sin(4)

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integral from 0 to 2 of 4xcos(x^2)

\int_{0}^{2} 4 x cos (x^{2}) d x

2 sin (4)

Decimal

- 1.51360 \dots

Solution steps

\int_{0}^{2} 4 x cos (x^{2}) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{2} x cos (x^{2}) d x

Apply u-substitution

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

= 4 \cdot \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

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

Use the common integral:

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

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

Simplify

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

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

Compute the boundaries:

sin (4)

= 2 sin (4)

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