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Identical expressions
sin^2x+4e^x
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sin^2x+4e^xdx
Similar expressions
sin^2x-4e^x

Integral of sin^2x+4e^x dx

The solution

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  0                    
  /                    
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 |  /   2         x\   
 |  \sin (x) + 4*E / dx
 |                     
/                      
0

$$\int\limits_{0}^{0} \left(4 e^{x} + \sin^{2}{\left(x \right)}\right)\, dx$$

Integral(sin(x)^2 + 4*E^x, (x, 0, 0))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 e^{x}\, dx = 4 \int e^{x}\, dx$
  1. The integral of the exponential function is itself.
    
    $\int e^{x}\, dx = e^{x}$
  So, the result is: $4 e^{x}$
1. Rewrite the integrand:
  
  $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
  The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
The result is: $\frac{x}{2} + 4 e^{x} - \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

$\frac{x}{2} + 4 e^{x} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} + 4 e^{x} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                              
 | /   2         x\          x      x   sin(2*x)
 | \sin (x) + 4*E / dx = C + - + 4*e  - --------
 |                           2             4    
/

$$\int \left(4 e^{x} + \sin^{2}{\left(x \right)}\right)\, dx = C + \frac{x}{2} + 4 e^{x} - \frac{\sin{\left(2 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

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Integral of sin^2x+4e^x dx

Limits of integration:

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Piecewise:

The solution