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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of x×e^x
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Integral of 1/(x²+1)
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Integral of 1/√(1-4x^2)
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Integral of x^2e^x
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Identical expressions
- sqrt(x)/(two +x^ two)
- square root of (x) divide by (2 plus x squared )
- square root of (x) divide by (two plus x to the power of two)
- √(x)/(2+x^2)
- sqrt(x)/(2+x2)
- sqrtx/2+x2
- sqrt(x)/(2+x²)
- sqrt(x)/(2+x to the power of 2)
- sqrtx/2+x^2
- sqrt(x) divide by (2+x^2)
- sqrt(x)/(2+x^2)dx
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Similar expressions
- sqrt(x)/(2-x^2)
Integral of sqrt(x)/(2+x^2) dx
The solution
You have entered
[src]
___ \/ 2 / | | ___ | \/ x | ------ dx | 2 | 2 + x | / 1
$$\int\limits_{1}^{\sqrt{2}} \frac{\sqrt{x}}{x^{2} + 2}\, dx$$
Integral(sqrt(x)/(2 + x^2), (x, 1, sqrt(2)))
The answer (Indefinite)
[src]
/ | | ___ 4 ___ / 4 ___ ___\ 4 ___ / 4 ___ ___\ 4 ___ / ___ 3/4 ___\ 4 ___ / ___ 3/4 ___\ | \/ x \/ 2 *atan\1 + \/ 2 *\/ x / \/ 2 *atan\-1 + \/ 2 *\/ x / \/ 2 *log\4*x + 4*\/ 2 + 4*2 *\/ x / \/ 2 *log\4*x + 4*\/ 2 - 4*2 *\/ x / | ------ dx = C + --------------------------- + ---------------------------- - --------------------------------------- + --------------------------------------- | 2 2 2 4 4 | 2 + x | /
$$2\,\left(-{{\log \left(x+2^{{{3}\over{4}}}\,\sqrt{x}+\sqrt{2}
\right)}\over{2^{{{11}\over{4}}}}}+{{\log \left(x-2^{{{3}\over{4}}}
\,\sqrt{x}+\sqrt{2}\right)}\over{2^{{{11}\over{4}}}}}+{{\arctan
\left({{2\,\sqrt{x}+2^{{{3}\over{4}}}}\over{2^{{{3}\over{4}}}}}
\right)}\over{2^{{{7}\over{4}}}}}+{{\arctan \left({{2\,\sqrt{x}-2^{
{{3}\over{4}}}}\over{2^{{{3}\over{4}}}}}\right)}\over{2^{{{7}\over{4
}}}}}\right)$$
The graph
The answer
[src]
4 ___ / 4 ___\ 4 ___ / 4 ___\ 4 ___ / ___\ 4 ___ / 3/4 ___\ 4 ___ 4 ___ / ___\ 4 ___ / ___ 3/4\
\/ 2 *atan\1 - \/ 2 / \/ 2 *atan\1 + \/ 2 / \/ 2 *log\8 + 8*\/ 2 / \/ 2 *log\4 - 4*2 + 4*\/ 2 / pi*\/ 2 \/ 2 *log\-8 + 8*\/ 2 / \/ 2 *log\4 + 4*\/ 2 + 4*2 /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
2 2 4 4 4 4 4
$$-{{\log \left(2^{{{3}\over{2}}}+2\right)}\over{2^{{{7}\over{4}}}}}+
{{\log \left(2^{{{3}\over{2}}}-2\right)}\over{2^{{{7}\over{4}}}}}+{{
\log \left(2^{{{3}\over{4}}}+\sqrt{2}+1\right)}\over{2^{{{7}\over{4
}}}}}-{{\log \left(-2^{{{3}\over{4}}}+\sqrt{2}+1\right)}\over{2^{{{7
}\over{4}}}}}-{{\arctan \left(2^{{{1}\over{4}}}+1\right)}\over{2^{{{
3}\over{4}}}}}-{{\arctan \left(2^{{{1}\over{4}}}-1\right)}\over{2^{
{{3}\over{4}}}}}+{{\pi}\over{2^{{{7}\over{4}}}}}$$
=
=
4 ___ / 4 ___\ 4 ___ / 4 ___\ 4 ___ / ___\ 4 ___ / 3/4 ___\ 4 ___ 4 ___ / ___\ 4 ___ / ___ 3/4\
\/ 2 *atan\1 - \/ 2 / \/ 2 *atan\1 + \/ 2 / \/ 2 *log\8 + 8*\/ 2 / \/ 2 *log\4 - 4*2 + 4*\/ 2 / pi*\/ 2 \/ 2 *log\-8 + 8*\/ 2 / \/ 2 *log\4 + 4*\/ 2 + 4*2 /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
2 2 4 4 4 4 4
$$- \frac{\sqrt[4]{2} \log{\left(8 + 8 \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 + \sqrt[4]{2} \right)}}{2} - \frac{\sqrt[4]{2} \log{\left(- 4 \cdot 2^{\frac{3}{4}} + 4 + 4 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 - \sqrt[4]{2} \right)}}{2} + \frac{\sqrt[4]{2} \log{\left(-8 + 8 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \log{\left(4 + 4 \sqrt{2} + 4 \cdot 2^{\frac{3}{4}} \right)}}{4} + \frac{\sqrt[4]{2} \pi}{4}$$
The graph
Use the examples entering the upper and lower limits of integration.