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Identical expressions
x^ three + two ^x+3sinx+3cosx
x cubed plus 2 to the power of x plus 3 sinus of x plus 3 co sinus of e of x
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x^3+2^x+3sinx+3cosxdx
Similar expressions
x^3+2^x-3sinx+3cosx
x^3+2^x+3sinx-3cosx
x^3-2^x+3sinx+3cosx

Solving integrals/ x^3+2^x+3sinx+3cosx

Integral of x^3+2^x+3sinx+3cosx dx

The solution

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  1                                   
  /                                   
 |                                    
 |  / 3    x                      \   
 |  \x  + 2  + 3*sin(x) + 3*cos(x)/ dx
 |                                    
/                                     
0

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

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

Detail solution

Integrate term-by-term:
1. The integral of an exponential function is itself divided by the natural logarithm of the base.
  
  $\int 2^{x}\, dx = \frac{2^{x}}{\log{\left(2 \right)}}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{3}\, dx = \frac{x^{4}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $- 3 \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \cos{\left(x \right)}\, dx = 3 \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $3 \sin{\left(x \right)}$
The result is: $\frac{2^{x}}{\log{\left(2 \right)}} + \frac{x^{4}}{4} + 3 \sin{\left(x \right)} - 3 \cos{\left(x \right)}$
Now simplify:

$\frac{2^{x}}{\log{\left(2 \right)}} + \frac{x^{4}}{4} - 3 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}$
Add the constant of integration:

$\frac{2^{x}}{\log{\left(2 \right)}} + \frac{x^{4}}{4} - 3 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$

The answer is:

$\frac{2^{x}}{\log{\left(2 \right)}} + \frac{x^{4}}{4} - 3 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                          
 |                                                                 4      x  
 | / 3    x                      \                                x      2   
 | \x  + 2  + 3*sin(x) + 3*cos(x)/ dx = C - 3*cos(x) + 3*sin(x) + -- + ------
 |                                                                4    log(2)
/

$$3\,\sin x-3\,\cos x+{{2^{x}}\over{\log 2}}+{{x^4}\over{4}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

13     1                         
-- + ------ - 3*cos(1) + 3*sin(1)
4    log(2)

$${{\left(12\,\sin 1-12\,\cos 1+13\right)\,\log 2+4}\over{4\,\log 2}}$$

13     1                         
-- + ------ - 3*cos(1) + 3*sin(1)
4    log(2)

$$- 3 \cos{\left(1 \right)} + \frac{1}{\log{\left(2 \right)}} + 3 \sin{\left(1 \right)} + \frac{13}{4}$$

Simplify

Numerical answer [src]

5.59620107770823

5.59620107770823

The graph

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

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Integral of x^3+2^x+3sinx+3cosx dx

Limits of integration:

The graph:

Piecewise:

The solution