Question: "Evaluate the integrals ∫14(1+u)12udu"

floymdiT

Answered question

2021-10-10

Evaluate the integrals

\int_{1}^{4} \frac{{(1 + \sqrt{u})}^{\frac{1}{2}}}{\sqrt{u}} d u

funblogC

Skilled2021-10-11Added 91 answers

Let

I = \int_{1}^{4} \frac{{(1 + \sqrt{u})}^{\frac{1}{2}}}{\sqrt{u}} d u

Solve above definite integral.

I = \int_{1}^{4} \frac{{(1 + \sqrt{u})}^{\frac{1}{2}}}{}} {\sqrt{u}} d u

Assume

a^{2} = 1 + \sqrt{u}

Derivative it with respect to u.

2 a \frac{d a}{d u} = 0 + \frac{1}{2 \sqrt{u}}

4 a d a = \frac{d u}{\sqrt{u}}

Replace

1 + \sqrt{u}

with

a^{2}

and

\frac{1}{\sqrt{u}} d u

with 4ada.

I = \int_{1}^{4} a 4 a d a

I = \int_{1}^{4} 4 a^{2} d a

I = 4 {[\frac{a^{3}}{3}]}_{1}^{4}

Put the value of a in above integral and solve the integral.

I = 4 {[\frac{{(\sqrt{1 + \sqrt{u}})}^{3}}{3}]}_{1}^{4}

I = 4 [\frac{{(\sqrt{3})}^{3}}{3} - \frac{{(\sqrt{2})}^{3}}{3}]

I = 4 [\sqrt{3} - \frac{2 \sqrt{2}}{3}]

I = \frac{4}{3} (3 \sqrt{3} - 2 \sqrt{2})

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