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

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

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

\int_{0}^{π} 2 x cos (x^{2}) d x

sin (π^{2})

Decimal

- 0.43030 \dots

Solution steps

\int_{0}^{π} 2 x cos (x^{2}) d x

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

= [sin (u)]_{0}^{π^{2}}

Compute the boundaries:

sin (π^{2})

= sin (π^{2})

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