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Identical expressions
x^ three *sqrt(one +x^ two)
x cubed multiply by square root of (1 plus x squared )
x to the power of three multiply by square root of (one plus x to the power of two)
x^3*√(1+x^2)
x3*sqrt(1+x2)
x3*sqrt1+x2
x³*sqrt(1+x²)
x to the power of 3*sqrt(1+x to the power of 2)
x^3sqrt(1+x^2)
x3sqrt(1+x2)
x3sqrt1+x2
x^3sqrt1+x^2
x^3*sqrt(1+x^2)dx
Similar expressions
x^3*sqrt(1-x^2)

Solving integrals/ x^3*sqrt(1+x^2)

Integral of x^3*sqrt(1+x^2) dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |        ________   
 |   3   /      2    
 |  x *\/  1 + x   dx
 |                   
/                    
0

$$\int\limits_{0}^{1} x^{3} \sqrt{x^{2} + 1}\, dx$$

Integral(x^3*sqrt(1 + x^2), (x, 0, 1))

Detail solution

Rewrite the integrand:

$x^{3} \sqrt{x^{2} + 1} = \frac{x^{5} + x^{3}}{\sqrt{x^{2} + 1}}$

SqrtQuadraticDenomRule(a=1, b=0, c=1, coeffs=[1, 0, 1, 0, 0, 0], context=(x**5 + x**3)/sqrt(x**2 + 1), symbol=x)
Now simplify:

$\frac{\sqrt{x^{2} + 1} \cdot \left(3 x^{4} + x^{2} - 2\right)}{15}$
Add the constant of integration:

$\frac{\sqrt{x^{2} + 1} \cdot \left(3 x^{4} + x^{2} - 2\right)}{15}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{x^{2} + 1} \cdot \left(3 x^{4} + x^{2} - 2\right)}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                                     
 |       ________             ________ /        4    2\
 |  3   /      2             /      2  |  2    x    x |
 | x *\/  1 + x   dx = C + \/  1 + x  *|- -- + -- + --|
 |                                     \  15   5    15/
/

$${{x^2\,\left(x^2+1\right)^{{{3}\over{2}}}}\over{5}}-{{2\,\left(x^2+ 1\right)^{{{3}\over{2}}}}\over{15}}$$

Simplify

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Plot the graph f(x)

The answer [src]

         ___
2    2*\/ 2 
-- + -------
15      15

$${{2^{{{3}\over{2}}}}\over{15}}+{{2}\over{15}}$$

         ___
2    2*\/ 2 
-- + -------
15      15

$$\frac{2}{15} + \frac{2 \sqrt{2}}{15}$$

Упростить

Numerical answer [src]

0.321895141649746

0.321895141649746

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of x^3*sqrt(1+x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution