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

The integral from 0 to 1 of (e^x+5)/(e^x+5x) is ln(e+5)

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integral from 0 to 1 of (e^x+5)/(e^x+5x)

\int_{0}^{1} \frac{e ^{x} + 5}{e ^{x} + 5 x} d x

ln (e + 5)

Decimal

2.04359 \dots

Solution steps

\int_{0}^{1} \frac{e ^{x} + 5}{e ^{x} + 5 x} d x

Apply u-substitution

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

= \int_{1}^{e + 5} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{e + 5}

Compute the boundaries:

ln (e + 5)

= ln (e + 5)

\int_{0}^{5} - x + 5 d x

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

\int_{- 4}^{5} x d x

\int_{4}^{9} \frac{1}{x} d x

\int_{0}^{4} (8 x - 32) d x

What is the integral from 0 to 1 of (e^x+5)/(e^x+5x) ?
The integral from 0 to 1 of (e^x+5)/(e^x+5x) is ln(e+5)