Question

∫cos⁡2x+2 sin²x /(cos²x) dx is equal to

Answer 1

Answer :

tan²x + C

To integrate :

$\sf \bullet \: \: \int \dfrac{\cos2x + 2\sin^{2}x}{\cos^{2} x} dx$

Formulae to be used :

$\sf \bullet \: \: \int \sec^{2} x dx = \tan x + C$

where C is a constant .

$\sf \bullet \: \: \cos2x = 1-\sin^{2}x = 2\cos^{2}x- 1 = \cos^{2}x - \sin^{2}x$

Solution :

$\sf \dashrightarrow I = \int \dfrac{\cos2x + 2\sin^{2}x}{\cos^{2}x} dx \\\\ \sf \dashrightarrow I = \int \dfrac{1-2\sin^{2}x + 2\sin^{2}x}{\cos^{2}x} dx \\\\ \sf \dashrightarrow I = \int \dfrac{1}{\cos^{2}x} dx\\\\ \sf \dashrightarrow I = \int\sec^{2} x dx \\\\ \sf \dashrightarrow I = \tan^{2}x + C$

Some other Identities :

$\sf \bullet \: \: \int x^{n} dx = \dfrac{x^{n+1}}{n+1} \\\\ \sf \bullet \: \: \int \dfrac{1}{x}dx = \log_{e} x + C \\\\ \sf \bullet \: \: \int \cos x dx = \sin x + C \\\\ \sf \bullet \: \: \int \sin x dx = -\cos x + C \\\\ \sf \bullet \: \: \int \csc^{2} x dx = -\cot x + C \\\\ \sf \bullet \: \: \int \sec x \tan x dx = \sec x + C \\\\ \sf \bullet \: \: \int \csc x \cot x dx = -\cosec x + C \\\\ \sf \bullet \: \: \int e^{x}dx = e^{x} + C \\\\ \sf \bullet \: \: \int a^{x} dx = \dfrac{a^{x}}{log_{e}^{a} } + C$