What is the integral from 0 to pi of 3x^2sin(x^3) ?

The integral from 0 to pi of 3x^2sin(x^3) is -cos(pi^3)+1

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integral from 0 to pi of 3x^2sin(x^3)

\int_{0}^{π} 3 x^{2} sin (x^{3}) d x

- cos (π^{3}) + 1

Solution steps

\int_{0}^{π} 3 x^{2} sin (x^{3}) d x

Take the constant out:

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

= 3 \cdot \int_{0}^{π} x^{2} sin (x^{3}) d x

Apply u-substitution

: \int_{0}^{π^{3}} \frac{sin ( u )}{3} d u

= 3 \cdot \int_{0}^{π^{3}} \frac{sin ( u )}{3} d u

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

3 \cdot \frac{1}{3} [- cos (u)]_{0}^{π^{3}} : [- cos (u)]_{0}^{π^{3}}

= [- cos (u)]_{0}^{π^{3}}

Compute the boundaries:

- cos (π^{3}) + 1

= - cos (π^{3}) + 1

\int_{0}^{\frac{π}{2}} e^{x} + cos (x) d x

\int_{0}^{z} e^{t^{2}} d t

\int_{0}^{2} 2 π (x - 3) x^{2} d x

\int_{- 1}^{1} (x^{3} - 3 x^{2} - x + 3) d x

\int_{0}^{1} \frac{2}{x ^{2}} d x

What is the integral from 0 to pi of 3x^2sin(x^3) ?
The integral from 0 to pi of 3x^2sin(x^3) is -cos(pi^3)+1