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

The integral from 1 to 5 of x/(sqrt(2x+1)) is (4sqrt(11))/3

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

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

\frac{4 11}{3}

Decimal

4.42216 \dots

Solution steps

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

Apply u-substitution

= \int_{3}^{11} \frac{u ^{2} - 1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{3}^{11} 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

= \frac{1}{2} (\int_{3}^{11} u^{2} d u - \int_{3}^{11} 1 d u)

\int_{3}^{11} u^{2} d u = \frac{11 11}{3} - 3

\int_{3}^{11} 1 d u = 11 - 3

= \frac{1}{2} (\frac{11 11}{3} - 3 - (11 - 3))

Simplify

\frac{1}{2} (\frac{11 11}{3} - 3 - (11 - 3)) : \frac{4 11}{3}

= \frac{4 11}{3}

What is the integral from 1 to 5 of x/(sqrt(2x+1)) ?
The integral from 1 to 5 of x/(sqrt(2x+1)) is (4sqrt(11))/3