What is the integral from 0 to 1 of 7x^6cos(x^7) ?

The integral from 0 to 1 of 7x^6cos(x^7) is sin(1)

integral from 0 to 1 of 7x^6cos(x^7)

\int_{0}^{1} 7 x^{6} cos (x^{7}) d x

sin (1)

Decimal

0.84147 \dots

Solution steps

\int_{0}^{1} 7 x^{6} cos (x^{7}) d x

Take the constant out:

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

= 7 \cdot \int_{0}^{1} x^{6} cos (x^{7}) d x

Apply u-substitution

= 7 \cdot \int_{0}^{1} \frac{cos ( u )}{7} d u

Take the constant out:

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

= 7 \cdot \frac{1}{7} \cdot \int_{0}^{1} cos (u) d u

Use the common integral:

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

= 7 \cdot \frac{1}{7} [sin (u)]_{0}^{1}

Simplify

7 \cdot \frac{1}{7} [sin (u)]_{0}^{1} : [sin (u)]_{0}^{1}

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

Compute the boundaries:

sin (1)

= sin (1)

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What is the integral from 0 to 1 of 7x^6cos(x^7) ?
The integral from 0 to 1 of 7x^6cos(x^7) is sin(1)