"Evaluate the integral. ∫x1+x2dx" was answered by our experts

he298c

Answered question

2021-10-24

Evaluate the integral.

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

i1ziZ

Skilled2021-10-25Added 92 answers

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

substitute

1 + x^{2} = t

differentiate both sides:
2xdx=dt

x d x = \frac{d t}{2}

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

= \frac{1}{2} \int t^{- \frac{1}{2}} d t

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

= \frac{1}{\neg {2}} (\frac{t^{\frac{1}{2}}}{\frac{1}{\neg {2}}}) + C

= t^{\frac{1}{2}} + C

since

t = 1 + x^{2}

= \sqrt{1 + x^{2}} + C

Answer:

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

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