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Identical expressions
1 two t^2×sqrt(t^ three + five)
12t squared × square root of (t cubed plus 5)
1 two t squared × square root of (t to the power of three plus five)
12t^2×√(t^3+5)
12t2×sqrt(t3+5)
12t2×sqrtt3+5
12t²×sqrt(t³+5)
12t to the power of 2×sqrt(t to the power of 3+5)
12t^2×sqrtt^3+5
Similar expressions
12t^2×sqrt(t^3-5)

Integral of 12t^2×sqrt(t^3+5) dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |           ________   
 |      2   /  3        
 |  12*t *\/  t  + 5  dt
 |                      
/                       
0

$$\int\limits_{0}^{1} 12 t^{2} \sqrt{t^{3} + 5}\, dt$$

Integral((12*t^2)*sqrt(t^3 + 5), (t, 0, 1))

Detail solution

Let $u = t^{3} + 5$ .

Then let $du = 3 t^{2} dt$ and substitute $4 du$ :

$\int 4 \sqrt{u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = 4 \int \sqrt{u}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $\frac{8 u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$\frac{8 \left(t^{3} + 5\right)^{\frac{3}{2}}}{3}$
Now simplify:

$\frac{8 \left(t^{3} + 5\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$\frac{8 \left(t^{3} + 5\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$\frac{8 \left(t^{3} + 5\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                                      3/2
 |          ________            / 3    \   
 |     2   /  3               8*\t  + 5/   
 | 12*t *\/  t  + 5  dt = C + -------------
 |                                  3      
/

$$\int 12 t^{2} \sqrt{t^{3} + 5}\, dt = C + \frac{8 \left(t^{3} + 5\right)^{\frac{3}{2}}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                ___
     ___   40*\/ 5 
16*\/ 6  - --------
              3

$$- \frac{40 \sqrt{5}}{3} + 16 \sqrt{6}$$

                ___
     ___   40*\/ 5 
16*\/ 6  - --------
              3

$$- \frac{40 \sqrt{5}}{3} + 16 \sqrt{6}$$

16*sqrt(6) - 40*sqrt(5)/3

Numerical answer [src]

9.37759618453365

9.37759618453365

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

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Identical expressions

Similar expressions

Integral of 12t^2×sqrt(t^3+5) dx

Limits of integration:

The graph:

Piecewise:

The solution