Solved: "Evaluate the following integrals. ∫1x+xdx"

he298c

Answered question

2021-10-20

Evaluate the following integrals.

\int \frac{1}{x + \sqrt{x}} d x

toroztatG

Skilled2021-10-21Added 98 answers

Step 1
Consider the integral:

\int \frac{1}{x + \sqrt{x}} d x

Apply u−substitution:

u = \sqrt{x}

\int \frac{1}{x + \sqrt{x}} d x = \int \frac{2}{u + 1} d u

Take the constant out:

\int \frac{2}{u + 1} d u = 2 \cdot \int \frac{1}{u + 1} d u

Let v=u+1

= 2 \cdot \int \frac{1}{v} d v

= 2 \ln | v | + C

= 2 \ln | u + 1 | + C

= 2 \ln | \sqrt{x} + 1 | + C

Step 2
Hence:

\int \frac{1}{x + \sqrt{x}} d x = 2 \ln | \sqrt{x} + 1 | + C

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