Question

$integrting \: byparts$
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Answer 1

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Answer 2

Hey Friend!

Here is your answer:

To find → ∫ x·sec²x dx

Few formulas to be applied here are:

1. ∫ u v dx = u ∫ v dx - ∫ u' (∫ v dx) dx      where u' is the first order derivative of u
2. ∫ xⁿ dx = $\frac{x^{n + 1}}{n + 1}$ + c                   where c is any constant
3. ∫ sec²x dx = tanx + c             where c is any constant
4. ∫ tan x = - ln(| cos x |) + c       where c is any constant
5. If y = xⁿ, then $\frac{dy}{dx}$ = nxⁿ⁻¹

Step by step solved answer:

Firstly, we have to find the first order derivative of u, ie u'

u' = $\frac{d}{dx}$x = 1 × x¹⁻¹ = 1

Now,

∫ x·sec²x dx = x ∫ sec²x dx - ∫ 1 (∫ sec²x dx) dx  

                  = x · tan x - ∫ 1 (tan x) dx

                  = x tan x - 1 ∫ tan x dx

                  = x tan x - ( - ln | cos x |)

                  = x tan x + ln | cos x | + c

Answer: x tan x + ln | cos x | + c      ∨ c → constant

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