Integral of xsec²(x) dx

You have entered [src]

  1             
  /             
 |              
 |       2      
 |  x*sec (x) dx
 |              
/               
0

$$\int\limits_{0}^{1} x \sec^{2}{\left(x \right)}\, dx$$

Integral(x*sec(x)^2, (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \sec^{2}{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. $\int \sec^{2}{\left(x \right)}\, dx = \tan{\left(x \right)}$
Now evaluate the sub-integral.
Rewrite the integrand:

$\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
Let $u = \cos{\left(x \right)}$ .

Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :

$\int \frac{1}{u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $- \log{\left(u \right)}$
Now substitute $u$ back in:

$- \log{\left(\cos{\left(x \right)} \right)}$
Add the constant of integration:

$x \tan{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$x \tan{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 |      2                                   
 | x*sec (x) dx = C + x*tan(x) + log(cos(x))
 |                                          
/

$${{\left(\sin ^2\left(2\,x\right)+\cos ^2\left(2\,x\right)+2\,\cos \left(2\,x\right)+1\right)\,\log \left(\sin ^2\left(2\,x\right)+ \cos ^2\left(2\,x\right)+2\,\cos \left(2\,x\right)+1\right)+4\,x\, \sin \left(2\,x\right)}\over{2\,\sin ^2\left(2\,x\right)+2\,\cos ^2 \left(2\,x\right)+4\,\cos \left(2\,x\right)+2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(cos(1)) + tan(1)

$${{\left(\sin ^22+\cos ^22+2\,\cos 2+1\right)\,\log \left(\sin ^22+ \cos ^22+2\,\cos 2+1\right)+4\,\sin 2}\over{2\,\sin ^22+2\,\cos ^22+ 4\,\cos 2+2}}-{{\log 4}\over{2}}$$

=

log(cos(1)) + tan(1)

$$\log{\left(\cos{\left(1 \right)} \right)} + \tan{\left(1 \right)}$$

Simplify

Numerical answer [src]

0.941781254268888

0.941781254268888

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xsec²(x) dx

Limits of integration:

The graph:

Piecewise:

The solution