Question

solve it fast for brainlist​

Answer 1

TO FIND :–

$\\ \implies \sf \int \dfrac{ { \sec}^{2}(x).dx }{9 - \tan(x) } = ?$

SOLUTION :–

• Let –

$\\ \implies \sf P = \int \dfrac{ { \sec}^{2}(x).dx}{9 - \tan(x) }$

• Now put tan(x) = t –

• Differentiate with respect to 't' –

$\\ \implies \sf \dfrac{d \{\tan(x) \} }{dt} = \dfrac{d(t)}{dt}$

$\\ \implies \sf \dfrac{d \{\tan(x) \} }{dt} =1$

$\\ \implies \sf { \sec}^{2} (x) \dfrac{dx}{dt} = 1$

$\\ \implies \sf { \sec}^{2} (x)dx = dt$

• So that –

$\\ \implies \sf P = \int \dfrac{dt}{9 -t }$

$\\ \implies \sf P =( - 1) \int \dfrac{dt}{t - 9}$

• Using identity –

$\\ \large\implies{ \boxed{\sf \int \dfrac{dx}{x} = \ln|x|}}$

• So that –

$\\ \implies \sf P =( - 1) \ln|{t - 9}|+c$

$\\ \implies \sf P = \ln|\bigg( \dfrac{1}{t - 9} \bigg)|+ c$

• Now replace 't' –

$\\ \large\implies{ \boxed{ \sf P = \ln \bigg(|\dfrac{1}{ \tan x- 9} | \bigg) + c}}$

Answer 2

☘️ See the attachment picture for Explanation.

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