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

The integral from 0 to 1 of (5t^4)/(1+t^5) is ln(2)

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

\int_{0}^{1} \frac{5 t ^{4}}{1 + t ^{5}} d t

ln (2)

Decimal Notation

0.69314 \dots

Solution steps

\int_{0}^{1} \frac{5 t ^{4}}{1 + 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} \frac{t ^{4}}{1 + t ^{5}} d t

Apply u-substitution

: \int_{1}^{2} \frac{1}{5 u} d u

= 5 \cdot \int_{1}^{2} \frac{1}{5 u} 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_{1}^{2} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{2}

Simplify

5 \cdot \frac{1}{5} [ln ∣ u ∣]_{1}^{2} : [ln ∣ u ∣]_{1}^{2}

= [ln ∣ u ∣]_{1}^{2}

Compute the boundaries:

ln (2)

= ln (2)

\int \frac{1}{y ( y + 2 )} d y

- 2 sin (x)

\int \frac{1}{x ^{2} 1 + x ^{2}} d x

\int \frac{8 x}{2 x ^{2} + 1} d x

y = (x + 1)^{3}

What is the integral from 0 to 1 of (5t^4)/(1+t^5) ?
The integral from 0 to 1 of (5t^4)/(1+t^5) is ln(2)