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Identical expressions
(x)^ three *sin((x)^ four)
(x) cubed multiply by sinus of ((x) to the power of 4)
(x) to the power of three multiply by sinus of ((x) to the power of four)
(x)3*sin((x)4)
x3*sinx4
(x)³*sin((x)⁴)
(x) to the power of 3*sin((x) to the power of 4)
(x)^3sin((x)^4)
(x)3sin((x)4)
x3sinx4
x^3sinx^4
(x)^3*sin((x)^4)dx

Integral of (x)^3*sin((x)^4) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |   3    / 4\   
 |  x *sin\x / dx
 |               
/                
0

$$\int\limits_{0}^{1} x^{3} \sin{\left(x^{4} \right)}\, dx$$

Integral(x^3*sin(x^4), (x, 0, 1))

Detail solution

Let $u = x^{4}$ .

Then let $du = 4 x^{3} dx$ and substitute $\frac{du}{4}$ :

$\int \frac{\sin{\left(u \right)}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                           
 |                        / 4\
 |  3    / 4\          cos\x /
 | x *sin\x / dx = C - -------
 |                        4   
/

$$\int x^{3} \sin{\left(x^{4} \right)}\, dx = C - \frac{\cos{\left(x^{4} \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(1)
- - ------
4     4

$$\frac{1}{4} - \frac{\cos{\left(1 \right)}}{4}$$

1   cos(1)
- - ------
4     4

$$\frac{1}{4} - \frac{\cos{\left(1 \right)}}{4}$$

1/4 - cos(1)/4

Numerical answer [src]

0.114924423532965

0.114924423532965

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

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Identical expressions

Integral of (x)^3*sin((x)^4) dx

Limits of integration:

The graph:

Piecewise:

The solution