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Solving integrals/ dx/1+3cos^2x

Integral of dx/1+3cos^2x dx

The solution

You have entered [src]

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

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

Simplify

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \cos^{2}{\left(x \right)}\, dx = 3 \int \cos^{2}{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      1. Let $u = 2 x$ .
        
        Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{4}\, du$
        
        The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        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: $\frac{\sin{\left(2 x \right)}}{4}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
    The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
  So, the result is: $\frac{3 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1 \cdot 1^{-1}\, dx = x$
The result is: $\frac{5 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4}$
Add the constant of integration:

$\frac{5 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{5 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                            
 | /  1        2   \          3*sin(2*x)   5*x
 | |1*- + 3*cos (x)| dx = C + ---------- + ---
 | \  1            /              4         2 
 |                                            
/

$${{3\,\left({{\sin \left(2\,x\right)}\over{2}}+x\right)}\over{2}}+x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5   3*cos(1)*sin(1)
- + ---------------
2          2

$${{3\,\sin 2+10}\over{4}}$$

5   3*cos(1)*sin(1)
- + ---------------
2          2

$$\frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{5}{2}$$

Simplify

Numerical answer [src]

3.18197307011926

3.18197307011926

The graph

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

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Integral of dx/1+3cos^2x dx

Limits of integration:

The graph:

Piecewise:

The solution