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Identical expressions
x/cos^ two x^2
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x/cos^2x^2dx

Solving integrals/ x/cos^2x^2

Integral of x/cos^2x^2 dx

The solution

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 pi           
 --           
 2            
  /           
 |            
 |     x      
 |  ------- dx
 |     4      
 |  cos (x)   
 |            
/             
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{x}{\cos^{4}{\left(x \right)}}\, dx$$

Integral(x/cos(x)^4, (x, 0, pi/2))

The answer (Indefinite) [src]

  /                                    4/x\                                 /       /x\\                             /        /x\\                                2/x\                                 /       2/x\\                                 5/x\                                      /x\                            2/x\    /       2/x\\                    4/x\    /       /x\\                     4/x\    /        /x\\                    6/x\    /       2/x\\                    6/x\    /       /x\\                     6/x\    /        /x\\                             3/x\                            2/x\    /       /x\\                     2/x\    /        /x\\                    4/x\    /       2/x\\      
 |                                2*tan |-|                            2*log|1 + tan|-||                        2*log|-1 + tan|-||                           2*tan |-|                            2*log|1 + tan |-||                          6*x*tan |-|                               6*x*tan|-|                       6*tan |-|*log|1 + tan |-||               6*tan |-|*log|1 + tan|-||                6*tan |-|*log|-1 + tan|-||               2*tan |-|*log|1 + tan |-||               2*tan |-|*log|1 + tan|-||                2*tan |-|*log|-1 + tan|-||                      4*x*tan |-|                       6*tan |-|*log|1 + tan|-||                6*tan |-|*log|-1 + tan|-||               6*tan |-|*log|1 + tan |-||      
 |    x                                 \2/                                 \       \2//                             \        \2//                                 \2/                                 \        \2//                                  \2/                                      \2/                             \2/    \        \2//                     \2/    \       \2//                      \2/    \        \2//                     \2/    \        \2//                     \2/    \       \2//                      \2/    \        \2//                              \2/                             \2/    \       \2//                      \2/    \        \2//                     \2/    \        \2//      
 | ------- dx = C - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + --------------------------------------
 |    4                       4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\
 | cos (x)          -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|
 |                             \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/
/

$$\int \frac{x}{\cos^{4}{\left(x \right)}}\, dx = C - \frac{6 x \tan^{5}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{4 x \tan^{3}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3}$$

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

2.26845469233483e+48

2.26845469233483e+48

The graph

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

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How to use it?

Identical expressions

Integral of x/cos^2x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution