What is the integral from 0 to 1 of 6x^5e^{x^6} ?

The integral from 0 to 1 of 6x^5e^{x^6} is e-1

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integral from 0 to 1 of 6x^5e^{x^6}

\int_{0}^{1} 6 x^{5} e^{x^{6}} d x

e - 1

Decimal Notation

1.71828 \dots

Solution steps

\int_{0}^{1} 6 x^{5} e^{x^{6}} d x

Take the constant out:

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

= 6 \cdot \int_{0}^{1} x^{5} e^{x^{6}} d x

Apply u-substitution

: \int_{0}^{1} \frac{e ^{u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

= 6 \cdot \int_{0}^{1} \frac{e ^{u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

Take the constant out:

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

= 6 \cdot \frac{1}{5} \cdot \int_{0}^{1} e^{u^{\frac{6}{5}}} u^{\frac{1}{5}} d u

Apply u-substitution

: \int_{0}^{1} \frac{5 e ^{v}}{6} d v

= 6 \cdot \frac{1}{5} \cdot \int_{0}^{1} \frac{5 e ^{v}}{6} d v

Take the constant out:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \int_{0}^{1} e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} [e^{v}]_{0}^{1}

Simplify

6 \cdot \frac{1}{5} \cdot \frac{5}{6} [e^{v}]_{0}^{1} : [e^{v}]_{0}^{1}

= [e^{v}]_{0}^{1}

Compute the boundaries:

e - 1

= e - 1

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What is the integral from 0 to 1 of 6x^5e^{x^6} ?
The integral from 0 to 1 of 6x^5e^{x^6} is e-1