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x^3sqrt(1+9x^4)
Solving integrals/ x^3sqrt(1+9x^4)

Integral of x^3sqrt(1+9x^4) dx

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The solution

You have entered [src] 
  1                    
  /                    
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 |        __________   
 |   3   /        4    
 |  x *\/  1 + 9*x   dx
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∫01x39x4+1 dx\int\limits_{0}^{1} x^{3} \sqrt{9 x^{4} + 1}\, dx 
Integral(x^3*sqrt(1 + 9*x^4), (x, 0, 1))
Detail solution

  1. Let u=9x4+1u = 9 x^{4} + 1.

    Then let du=36x3dxdu = 36 x^{3} dx and substitute du36\frac{du}{36}:

    ∫u36 du\int \frac{\sqrt{u}}{36}\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫u du=∫u du36\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{36}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: u3254\frac{u^{\frac{3}{2}}}{54}

    Now substitute uu back in:

    (9x4+1)3254\frac{\left(9 x^{4} + 1\right)^{\frac{3}{2}}}{54}

  2. Add the constant of integration:

    (9x4+1)3254+constant\frac{\left(9 x^{4} + 1\right)^{\frac{3}{2}}}{54}+ \mathrm{constant}

The answer is:

(9x4+1)3254+constant\frac{\left(9 x^{4} + 1\right)^{\frac{3}{2}}}{54}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                       
 |                                     3/2
 |       __________          /       4\   
 |  3   /        4           \1 + 9*x /   
 | x *\/  1 + 9*x   dx = C + -------------
 |                                 54     
/
∫x39x4+1 dx=C+(9x4+1)3254\int x^{3} \sqrt{9 x^{4} + 1}\, dx = C + \frac{\left(9 x^{4} + 1\right)^{\frac{3}{2}}}{54}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
           ____
  1    5*\/ 10 
- -- + --------
  54      27
−154+51027- \frac{1}{54} + \frac{5 \sqrt{10}}{27} 
=
= 
           ____
  1    5*\/ 10 
- -- + --------
  54      27
−154+51027- \frac{1}{54} + \frac{5 \sqrt{10}}{27} 
-1/54 + 5*sqrt(10)/27
Numerical answer [src] 
0.567088455586737
0.567088455586737
The graph
Integral of x^3sqrt(1+9x^4) dx

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

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