What is the integral from 2 to 5 of x/(sqrt(x-1)) ?

The integral from 2 to 5 of x/(sqrt(x-1)) is 20/3

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integral from 2 to 5 of x/(sqrt(x-1))

\int_{2}^{5} \frac{x}{x - 1} d x

\frac{20}{3}

Decimal Notation

6.66666 \dots

Solution steps

\int_{2}^{5} \frac{x}{x - 1} d x

Apply u-substitution

: \int_{1}^{2} 2 (u^{2} + 1) d u

= \int_{1}^{2} 2 (u^{2} + 1) 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} u^{2} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int_{1}^{2} u^{2} d u + \int_{1}^{2} 1 d u)

\int_{1}^{2} u^{2} d u = \frac{7}{3}

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

= 2 (\frac{7}{3} + 1)

Simplify

2 (\frac{7}{3} + 1) : \frac{20}{3}

= \frac{20}{3}

\frac{d}{d x} (\frac{1}{4 x ^{2} + 1})

\frac{\partial}{\partial x} (\frac{( 2 x + 3 y )}{4 x + 5 y})

f (x) = x^{2} + 5

cos (π x), 1

\int_{0}^{1} \frac{x ^{4} + 1}{x} d x

What is the integral from 2 to 5 of x/(sqrt(x-1)) ?
The integral from 2 to 5 of x/(sqrt(x-1)) is 20/3