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Integral of d{x}:
Integral of sec
Integral of 2xdx
Integral of 1/t
Integral of √(1-x²)
Identical expressions
tanxsec^3x
tangent of xsec cubed x
tanxsec3x
tanxsec³x
tanxsec to the power of 3x
tanxsec^3xdx

Solving integrals/ tanxsec^3x

Integral of tanxsec^3x dx

The solution

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  1                  
  /                  
 |                   
 |            3      
 |  tan(x)*sec (x) dx
 |                   
/                    
0

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

Integral(tan(x)*sec(x)^3, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sec^{3}{\left(x \right)}$ .
  
  Then let $du = 3 \tan{\left(x \right)} \sec^{3}{\left(x \right)} dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{1}{9}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{3}\, du = \frac{\int 1\, du}{3}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $\frac{u}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sec^{3}{\left(x \right)}}{3}$
Method #2
1. Let $u = \sec{\left(x \right)}$ .
  
  Then let $du = \tan{\left(x \right)} \sec{\left(x \right)} dx$ and substitute $du$ :
  
  $\int u^{2}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{2}\, du = \frac{u^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sec^{3}{\left(x \right)}}{3}$
Add the constant of integration:

$\frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                            3   
 |           3             sec (x)
 | tan(x)*sec (x) dx = C + -------
 |                            3   
/

$$\int \tan{\left(x \right)} \sec^{3}{\left(x \right)}\, dx = C + \frac{\sec^{3}{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1       1    
- - + ---------
  3        3   
      3*cos (1)

$$- \frac{1}{3} + \frac{1}{3 \cos^{3}{\left(1 \right)}}$$

  1       1    
- - + ---------
  3        3   
      3*cos (1)

$$- \frac{1}{3} + \frac{1}{3 \cos^{3}{\left(1 \right)}}$$

Упростить

Numerical answer [src]

1.7800013582586

1.7800013582586

The graph

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

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Identical expressions

Integral of tanxsec^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2