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Similar expressions
2√x(x^3-3)

Integral of 2√x(x^3+3) dx

The solution

You have entered [src]

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 |  2*t*x*\x  + 3/  dx
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0

$$\int\limits_{0}^{1} 2 t x \left(x^{3} + 3\right)^{2}\, dx$$

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

Detail solution

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

$\int 2 t x \left(x^{3} + 3\right)^{2}\, dx = 2 t \int x \left(x^{3} + 3\right)^{2}\, dx$
1. Rewrite the integrand:
  
  $x \left(x^{3} + 3\right)^{2} = x^{7} + 6 x^{4} + 9 x$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{7}\, dx = \frac{x^{8}}{8}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 6 x^{4}\, dx = 6 \int x^{4}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
    So, the result is: $\frac{6 x^{5}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 9 x\, dx = 9 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{9 x^{2}}{2}$
  The result is: $\frac{x^{8}}{8} + \frac{6 x^{5}}{5} + \frac{9 x^{2}}{2}$
So, the result is: $2 t \left(\frac{x^{8}}{8} + \frac{6 x^{5}}{5} + \frac{9 x^{2}}{2}\right)$
Now simplify:

$\frac{t x^{2} \cdot \left(5 x^{6} + 48 x^{3} + 180\right)}{20}$
Add the constant of integration:

$\frac{t x^{2} \cdot \left(5 x^{6} + 48 x^{3} + 180\right)}{20}+ \mathrm{constant}$

The answer is:

$\frac{t x^{2} \cdot \left(5 x^{6} + 48 x^{3} + 180\right)}{20}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                                                
 |               2              / 8      5      2\
 |       / 3    \               |x    6*x    9*x |
 | 2*t*x*\x  + 3/  dx = C + 2*t*|-- + ---- + ----|
 |                              \8     5      2  /
/

$${{t\,\left(5\,x^8+48\,x^5+180\,x^2\right)}\over{20}}$$

Simplify

The answer [src]

233*t
-----
  20

$${{233\,t}\over{20}}$$

233*t
-----
  20

$$\frac{233 t}{20}$$

Simplify

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

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Integral of 2√x(x^3+3) dx

Limits of integration:

The graph:

Piecewise:

The solution