What is the integral from 0 to 5 of 12sqrt(t/(t+1)) ?

The integral from 0 to 5 of 12sqrt(t/(t+1)) is 12(-ln(sqrt(6)+sqrt(5))+sqrt(30))

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integral from 0 to 5 of 12sqrt(t/(t+1))

\int_{0}^{5} 12 \frac{t}{t + 1} d t

12 (- ln (6 + 5) + 30)

Decimal Notation

47.19288 \dots

Solution steps

\int_{0}^{5} 12 \frac{t}{t + 1} d t

Take the constant out:

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

= 12 \cdot \int_{0}^{5} \frac{t}{t + 1} d t

Apply u-substitution

: \int_{1}^{6} \frac{u - 1}{u} d u

= 12 \cdot \int_{1}^{6} \frac{u - 1}{u} d u

Apply Integration By Parts

: [\frac{u - 1}{u} u - \int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u]_{1}^{6}

= 12 [\frac{u - 1}{u} u - \int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u]_{1}^{6}

\int \frac{1}{2 u ^{\frac{1}{2}} u - 1} d u = ln u + u - 1

= 12 [\frac{u - 1}{u} u - ln u + u - 1]_{1}^{6}

Compute the boundaries:

- ln (6 + 5) + 30

= 12 (- ln (6 + 5) + 30)

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x sin (x)

\frac{\partial}{\partial x} (x + x^{2} + y^{2})

What is the integral from 0 to 5 of 12sqrt(t/(t+1)) ?
The integral from 0 to 5 of 12sqrt(t/(t+1)) is 12(-ln(sqrt(6)+sqrt(5))+sqrt(30))