Evaluate : ∫ (cos 2x + 2sin² x)/(cos² x) dx. – Class 12 Business Mathematics Statistics | ConceptFirst

Question 14 of 42

\int \dfrac{\cos 2x + 2\sin^2 x}{\cos^2 x}\, dx

Tamil Nadu DgeTamil Nadu HSC (DGE) Commerce Board 2020Subjective· 2mImportance★★★★★

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Use the identity $\cos 2x = 1 - 2\sin^2 x$ so the numerator becomes $1$ ; the integral collapses to $\int \sec^2 x\,dx = \tan x + C$ .

Step 1 — Simplify the numerator. Using $\cos 2x = 1 - 2\sin^2 x$ :

$\cos 2x + 2\sin^2 x = (1 - 2\sin^2 x) + 2\sin^2 x = 1.$

Step 2 — Rewrite the integrand.

$\frac{\cos 2x + 2\sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x.$

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2024Set MARCH1 markMCQ
$\int \frac{\sin 2x}{2\cos x}\,dx$
(a) $\cos x+c$
(b) $\sin x+c$
(c) $-\sin x+c$
(d) $-\cos x+c$
›Reveal solutionSolution

$\int\frac{\sin 2x}{2\cos x}\,dx=\int\sin x\,dx=-\cos x+c$ .

Using the identity $\sin 2x=2\sin x\cos x$ ,

…

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CBSE 2023Set MARCH1 markMCQ
$\int \dfrac{\sin 2x}{2\sin x}\,dx$
(a) $\cos x+c$
(b) $\sin x+c$
(c) $\dfrac{1}{2}\cos x+c$
(d) $\dfrac{1}{2}\sin x+c$
›Reveal solutionSolution

Simplify the integrand with the double-angle identity $\sin 2x=2\sin x\cos x$ ; it collapses to $\int\cos x\,dx=\sin x+c$ .

Start with the given integral:

$\int\frac{\sin 2x}{2\sin x}\,dx.$

Apply $\sin 2x=2\sin x\cos x$ : …

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CBSE 2023Set MARCH1 markMCQ
$\int \sqrt{e^x}\,dx$
(a) $\dfrac{1}{2}\sqrt{e^x}+c$
(b) $\sqrt{e^x}+c$
(c) $\dfrac{1}{2\sqrt{e^x}}+c$
(d) $2\sqrt{e^x}+c$
›Reveal solutionSolution

Convert the surd to an exponent, $\sqrt{e^x}=e^{x/2}$ , then use $\int e^{ax}dx=\frac{1}{a}e^{ax}$ with $a=\tfrac12$ .

Rewrite the integrand:

$\int\sqrt{e^x}\,dx=\int e^{x/2}\,dx.$

…

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