What is the integral from 0 to 3 of x/(1+sqrt(1+x)) ?

The integral from 0 to 3 of x/(1+sqrt(1+x)) is 5/3

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integral from 0 to 3 of x/(1+sqrt(1+x))

\int_{0}^{3} \frac{x}{1 + 1 + x} d x

\frac{5}{3}

Decimal

1.66666 \dots

Solution steps

\int_{0}^{3} \frac{x}{1 + 1 + x} d x

Apply u-substitution

: \int_{1}^{4} \frac{u - 1}{1 + u} d u

= \int_{1}^{4} \frac{u - 1}{1 + u} d u

\frac{u - 1}{1 + u} = u - 1

= \int_{1}^{4} u - 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

= \int_{1}^{4} u d u - \int_{1}^{4} 1 d u

\int_{1}^{4} u d u = \frac{14}{3}

\int_{1}^{4} 1 d u = 3

= \frac{14}{3} - 3

Simplify

\frac{14}{3} - 3 : \frac{5}{3}

= \frac{5}{3}

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What is the integral from 0 to 3 of x/(1+sqrt(1+x)) ?
The integral from 0 to 3 of x/(1+sqrt(1+x)) is 5/3