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Question: How do you integrate \(\int {x\,{{\sec }^2}\,x} \) by integration by parts method?...

How do you integrate ∫x sec⁡2 x\int {x\,{{\sec }^2}\,x} by integration by parts method?

Explanation

Solution

We have a question where we are supposed to integrate the function with the help of integration by parts method. It is like the product rule for integration. In fact, it is also derived from the product rule for differentiation.
The rule is:
∫udv=uv−∫vdu\int {udv = uv - \int {vdu} }

Complete step-by-step solution:
We have a function ∫x sec⁡2 x\int {x\,{{\sec }^2}\,x} , and to find the integration by parts, certain steps are to be followed,
This is a classic case of integration by parts, which takes the form
∫udv=uv−∫vdu\smallint udv = uv - \smallint vdu
For the given integral ∫x sec2(x)dx\smallint x\,se{c^2}(x)dx , we want to choose a value of uu that gets simpler when we differentiate it and a value of dvdv that is easily integrated. So, we will substitute the value of the given equation in the formula that we have,
So, let:
u=x,⇒du=dxu = x, \Rightarrow du = dx
v=tan⁡x,⇒dv=sec⁡2xv = \tan x, \Rightarrow dv = {\sec ^2}x
We then have,
⇒∫x sec⁡2(x)dx=uv−∫vdu\Rightarrow \smallint x\,{\sec ^2}(x)dx = uv - \smallint vdu
Putting the values in the formula,
⇒xtan⁡(x)−∫tan⁡(x)dx\Rightarrow x\tan (x) - \smallint \tan (x)dx
You may be aware of the integral of tan(x)  tan(x)\; . If not, it's easy to find:
⇒xtan(x)−∫sin⁡(x)cos⁡(x)dx\Rightarrow xtan(x) - \int {\dfrac{{\sin (x)}}{{\cos (x)}}} dx
Let t=cos(x)t = cos(x) , implying that dt=−sin(x)dxdt = - sin(x)dx
⇒∫xsec⁡2(x)dx=xtan⁡(x)+∫−sin⁡(x)cos⁡(x)dx\Rightarrow \smallint x{\sec ^2}(x)dx = x\tan (x) + \smallint \dfrac{{ - \sin (x)}}{{\cos (x)}}dx
Solving the equation, until it is simplified,
⇒∫xsec⁡2(x)dx=xtan⁡(x)+∫1tdt\Rightarrow \smallint x{\sec ^2}(x)dx = x\tan (x) + \smallint \dfrac{1}{t}dt
This is a common integral:
⇒∫xsec⁡2(x)dx=xtan⁡(x)+ln⁡∣t∣+C\Rightarrow \smallint x{\sec ^2}(x)dx = x\tan (x) + \ln \left| t \right| + C
Working back from t=cos(x)t = cos(x):
∫xsec⁡2(x)dx=xtan⁡(x)+ln⁡∣cos⁡(x)∣+C  \smallint x{\sec ^2}(x)dx = x\tan (x) + \ln |\cos (x)| + C\;

Therefore, for an equation ∫x sec⁡2 x\int {x\,{{\sec }^2}\,x} the integration by parts will make the solution come up to xtan⁡(x)+ln⁡∣cos⁡x∣+Cx\tan (x) + \ln |\cos x| + C

Note: Integration is a way of adding the slices to add the whole. It is the calculation of an integral. Integration by parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways. Integration by parts is basically used for functions that can also be constructed as products of some other function and a third function’s derivative.