Best solutions to - "Evaluate the integral. ∫xdx1+x2"

lilyta79jd

Answered question

2021-11-09

Evaluate the integral.

\int \frac{x d x}{\sqrt{1 + x^{2}}}

Geraldine Flores

Beginner2021-11-10Added 21 answers

Step 1
Consider the following integral:

\int \frac{x d x}{\sqrt{1 + x^{2}}}

Substitute

1 + x^{2} = u \to 2 x d x = d u \to x d x = \frac{1}{2} d u

in the above integral:

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

= \frac{u^{- \frac{1}{2} + 1}}{2 (- \frac{1}{2} + 1)} + C

= \frac{u^{\frac{1}{2}}}{2 \times \frac{1}{2}} + C

= \sqrt{u} + C

Step 2
Resubstitute

u = 1 + x^{2}

in the above equation:

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

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