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Identical expressions
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Similar expressions
5cos*7x-2^x

Integral of 5cos*7x+2^x dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |  /              x\   
 |  \5*cos(7*x) + 2 / dx
 |                      
/                       
0

$$\int\limits_{0}^{1} \left(2^{x} + 5 \cos{\left(7 x \right)}\right)\, dx$$

Integral(5*cos(7*x) + 2^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 a constant times a function is the constant times the integral of the function:
  
  $\int 5 \cos{\left(7 x \right)}\, dx = 5 \int \cos{\left(7 x \right)}\, dx$
  1. Let $u = 7 x$ .
    
    Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
    
    $\int \frac{\cos{\left(u \right)}}{7}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(7 x \right)}}{7}$
  So, the result is: $\frac{5 \sin{\left(7 x \right)}}{7}$
The result is: $\frac{2^{x}}{\log{\left(2 \right)}} + \frac{5 \sin{\left(7 x \right)}}{7}$
Now simplify:

$\frac{2^{x} + \frac{\log{\left(32 \right)} \sin{\left(7 x \right)}}{7}}{\log{\left(2 \right)}}$
Add the constant of integration:

$\frac{2^{x} + \frac{\log{\left(32 \right)} \sin{\left(7 x \right)}}{7}}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer is:

$\frac{2^{x} + \frac{\log{\left(32 \right)} \sin{\left(7 x \right)}}{7}}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                            x  
 | /              x\          5*sin(7*x)     2   
 | \5*cos(7*x) + 2 / dx = C + ---------- + ------
 |                                7        log(2)
/

$$\int \left(2^{x} + 5 \cos{\left(7 x \right)}\right)\, dx = \frac{2^{x}}{\log{\left(2 \right)}} + C + \frac{5 \sin{\left(7 x \right)}}{7}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1      5*sin(7)
------ + --------
log(2)      7

$$\frac{5 \sin{\left(7 \right)}}{7} + \frac{1}{\log{\left(2 \right)}}$$

  1      5*sin(7)
------ + --------
log(2)      7

$$\frac{5 \sin{\left(7 \right)}}{7} + \frac{1}{\log{\left(2 \right)}}$$

1/log(2) + 5*sin(7)/7

Numerical answer [src]

1.91197118283096

1.91197118283096

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

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Identical expressions

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Integral of 5cos*7x+2^x dx

Limits of integration:

The graph:

Piecewise:

The solution