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Integral of d{x}:
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Integral of t^3sin(2t^4)dt
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Identical expressions
t^3sin(2t^ four)dt
t cubed sinus of (2t to the power of 4)dt
t cubed sinus of (2t to the power of four)dt
t3sin(2t4)dt
t3sin2t4dt
t³sin(2t⁴)dt
t to the power of 3sin(2t to the power of 4)dt
t^3sin2t^4dt

Solving integrals/ t^3sin(2t^4)dt

Integral of t^3sin(2t^4)dt dx

The solution

You have entered [src]

  1                  
  /                  
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 |   3    /   4\     
 |  t *sin\2*t /*1 dt
 |                   
/                    
0

$$\int\limits_{0}^{1} t^{3} \sin{\left(2 t^{4} \right)} 1\, dt$$

Integral(t^3*sin(2*t^4)*1, (t, 0, 1))

Detail solution

Let $u = 2 t^{4}$ .

Then let $du = 8 t^{3} dt$ and substitute $\frac{du}{8}$ :

$\int \frac{\sin{\left(u \right)}}{64}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(u \right)}}{8}\, du = \frac{\int \sin{\left(u \right)}\, du}{8}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{8}$
Now substitute $u$ back in:

$- \frac{\cos{\left(2 t^{4} \right)}}{8}$
Add the constant of integration:

$- \frac{\cos{\left(2 t^{4} \right)}}{8}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(2 t^{4} \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                            /   4\
 |  3    /   4\            cos\2*t /
 | t *sin\2*t /*1 dt = C - ---------
 |                             8    
/

$$-{{\cos \left(2\,t^4\right)}\over{8}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(2)
- - ------
8     8

$${{1}\over{8}}-{{\cos 2}\over{8}}$$

1   cos(2)
- - ------
8     8

$$- \frac{\cos{\left(2 \right)}}{8} + \frac{1}{8}$$

Simplify

Numerical answer [src]

0.177018354568393

0.177018354568393

The graph

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

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Identical expressions

Integral of t^3sin(2t^4)dt dx

Limits of integration:

The graph:

Piecewise:

The solution