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Question: How do you evaluate the integral \(\int {x{{\sec }^2}xdx} \)?...

How do you evaluate the integral ∫xsec⁡2xdx\int {x{{\sec }^2}xdx} ?

Explanation

Solution

To solve this integral, you need to know the integration by parts method and the formula for that is ∫udv=uv−∫vdu\int {udv = uv - \int {vdu} } and we should now the basic formula such as∫sec⁡2xdx=tan⁡x\int {{{\sec }^2}xdx = \tan x} and tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}}.

Complete step by step answer:
Let us consider the given solution:
∫xsec⁡2xdx\int {x{{\sec }^2}xdx}
Let’s consider I=∫xsec⁡2xdxI = \int {x{{\sec }^2}xdx}
Let us apply ∫udv\int {udv} formula to solve this equation, for that let us consider u=xu = x and dv=sec⁡2xdxdv = {\sec ^2}xdx and the formula for integration by parts is∫udv=uv−∫vdu\int {udv = uv - \int {vdu} } , if you look at the formula it contains dudu and vv, to find it we have to differentiate uu and integrate dvdv we get,
u=x,du=dxu = x,du = dx and dv=sec⁡2xdxdv = {\sec ^2}xdx, v=tan⁡xv = \tan x, substituting the value in the formula we get,
⇒I=∫xsec⁡2x=xtan⁡x−∫tan⁡xdx ⇒I=xtan⁡x−∫sin⁡xcos⁡xdx ⇒I=xtan⁡x+∫−sin⁡xcos⁡xdx  \Rightarrow I = \int {x{{\sec }^2}x = x\tan x - \int {\tan x dx} } \\\ \Rightarrow I = x\tan x - \int {\dfrac{{\sin x}}{{\cos x}}dx} \\\ \Rightarrow I = x\tan x + \int {\dfrac{{ - \sin x}}{{\cos x}}dx} \\\
Let t=cos⁡xt = \cos x and the by differentiating tt we get,
⇒dt=−sin⁡xdx\Rightarrow dt = - \sin x dx, by substituting the values, we get,
⇒I=xtan⁡x+∫1tdt\Rightarrow I = x\tan x + \int {\dfrac{1}{t}dt}
⇒I=xtan⁡x+∫1tdt\Rightarrow I = x\tan x + \int {\dfrac{1}{t}dt}
⇒I=xtan⁡x+log⁡t+C\Rightarrow I = x\tan x + \log t + C
This is the required solution.

Note: In I=xtan⁡x+log⁡t+CI = x\tan x + \log t + C, where C is the constant, whenever we integrate a logarithm, we will get a constant to balance the LHS and RHS. The formula for tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}} and the cot⁡x\cot x will be the reciprocal of tan⁡x\tan x, which will be equal to cot⁡x=cos⁡xsin⁡x\cot x = \dfrac{{\cos x}}{{\sin x}}. And know the other identities of the trigonometry also. Because in many of the materials, they don’t solve every step, they will just skip some of the steps and proceed to the answer. So knowing the basic identities will help you to follow all the steps in your materials.
This is an indefinite integral, while solving a indefinite integral we should be more careful. Because this is the most complicated and important problem which will be continuously used in your higher studies. Never hesitate to solve more problems. Solving more problems will relieve you from difficulties.