What is the integral from 0 to 1 of 5t^4sin(t^5) ?

The integral from 0 to 1 of 5t^4sin(t^5) is -cos(1)+1

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integral from 0 to 1 of 5t^4sin(t^5)

\int_{0}^{1} 5 t^{4} sin (t^{5}) d t

- cos (1) + 1

Decimal

0.45969 \dots

Solution steps

\int_{0}^{1} 5 t^{4} sin (t^{5}) d t

Take the constant out:

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

= 5 \cdot \int_{0}^{1} t^{4} sin (t^{5}) d t

Apply u-substitution

: \int_{0}^{1} \frac{sin ( u )}{5} d u

= 5 \cdot \int_{0}^{1} \frac{sin ( u )}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int_{0}^{1} sin (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{5} [- cos (u)]_{0}^{1}

Simplify

5 \cdot \frac{1}{5} [- cos (u)]_{0}^{1} : [- cos (u)]_{0}^{1}

= [- cos (u)]_{0}^{1}

Compute the boundaries:

- cos (1) + 1

= - cos (1) + 1

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What is the integral from 0 to 1 of 5t^4sin(t^5) ?
The integral from 0 to 1 of 5t^4sin(t^5) is -cos(1)+1