What is the integral from 1 to 4 of 1/(x+2sqrt(x)) ?

The integral from 1 to 4 of 1/(x+2sqrt(x)) is 2(2ln(2)-ln(3))

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integral from 1 to 4 of 1/(x+2sqrt(x))

\int_{1}^{4} \frac{1}{x + 2 x} d x

2 (2 ln (2) - ln (3))

Decimal

0.57536 \dots

Solution steps

\int_{1}^{4} \frac{1}{x + 2 x} d x

Apply u-substitution

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

= \int_{1}^{2} \frac{2}{u + 2} d u

Take the constant out:

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

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

Apply u-substitution

: \int_{3}^{4} \frac{1}{v} d v

= 2 \cdot \int_{3}^{4} \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 [ln ∣ v ∣]_{3}^{4}

Compute the boundaries:

2 ln (2) - ln (3)

= 2 (2 ln (2) - ln (3))

\frac{d}{d x} (\frac{e ^{- x}}{2})

\frac{d}{d y} (x^{2} + y^{2} - 5)

g (w) = \frac{9}{8 w ^{5}} + 9 w

\int_{0}^{1} \frac{2}{5} x^{2} (x + 2) d x

\int \frac{1}{e ^{- 4 x}} d x

What is the integral from 1 to 4 of 1/(x+2sqrt(x)) ?
The integral from 1 to 4 of 1/(x+2sqrt(x)) is 2(2ln(2)-ln(3))