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three +tan(5x)
3 plus tangent of (5x)
three plus tangent of (5x)
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Similar expressions
3-tan(5x)

Solving integrals/ 3+tan(5x)

Integral of 3+tan(5x) dx

The solution

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 |  (3 + tan(5*x)) dx
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0

$$\int\limits_{0}^{1} \left(\tan{\left(5 x \right)} + 3\right)\, dx$$

Integral(3 + tan(5*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Rewrite the integrand:
  
  $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
2. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
    
    $\int \frac{1}{25 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{5 u}\right)\, du = - \frac{\int \frac{1}{u}\, du}{5}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \frac{\log{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
  Method #2
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{25 \cos{\left(u \right)}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{5 \cos{\left(u \right)}}\, du = \frac{\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du}{5}$
      
      Let $u = \cos{\left(u \right)}$ .
      
      Then let $du = - \sin{\left(u \right)} du$ and substitute $- du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \log{\left(\cos{\left(u \right)} \right)}$
      
      So, the result is: $- \frac{\log{\left(\cos{\left(u \right)} \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 3\, dx = 3 x$
The result is: $3 x - \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
Add the constant of integration:

$3 x - \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}+ \mathrm{constant}$

The answer is:

$3 x - \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                               log(cos(5*x))
 | (3 + tan(5*x)) dx = C + 3*x - -------------
 |                                     5      
/

$${{\log \sec \left(5\,x\right)}\over{5}}+3\,x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$-{{\log \cos 5-15}\over{5}}$$

nan

$$\text{NaN}$$

Simplify

Numerical answer [src]

4.33441279257591

4.33441279257591

The graph

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

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Integral of 3+tan(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2