Question

\(\int\frac{2\tan x+3}{\sin^2x+2\cos^2x}dx=\)

• A. \(\frac{3}{\sqrt2}sin^{-1}\frac{\sin x}{\sqrt2}+ln|\sin^2x+2|+C\)
• B. \(\frac{3}{\sqrt2}tan^{-1}\frac{\tan x}{\sqrt2}+ln|\tan^2x+2|+C\)
• C. \(\frac{3}{\sqrt2}tan^{-1}\frac{\tan x}{\sqrt2}-ln|\tan^2x+2|+C\)
• D. \(\frac{3}{\sqrt2}cos^{-1}\frac{\cos x}{\sqrt2}+ln|\sin^2x+2|+C\)
• E. \(\frac{3}{\sqrt2}cos^{-1}\frac{\cos x}{\sqrt2}-ln|\cos^2x+2|+C\)

Answer 1

The Correct Option is B

Step 1: Simplify the denominator: \[ \sin^2 x + 2\cos^2 x = \sin^2 x + \cos^2 x + \cos^2 x = 1 + \cos^2 x \]

Step 2: Rewrite the integral using trigonometric identities: \[ \int \frac{2\tan x + 3}{1 + \cos^2 x} dx \]

Step 3: Make the substitution \( u = \tan x \), \( du = \sec^2 x dx \), and note that: \[ \cos^2 x = \frac{1}{1 + \tan^2 x} = \frac{1}{1 + u^2} \] \[ 1 + \cos^2 x = 1 + \frac{1}{1 + u^2} = \frac{2 + u^2}{1 + u^2} \]

Step 4: Transform the integral: \[ \int \frac{2u + 3}{\frac{2 + u^2}{1 + u^2}} \cdot \frac{du}{1 + u^2} = \int \frac{(2u + 3)(1 + u^2)}{2 + u^2} \cdot \frac{du}{1 + u^2} = \int \frac{2u + 3}{2 + u^2} du \]

Step 5: Split the integral: \[ \int \frac{2u}{2 + u^2} du + \int \frac{3}{2 + u^2} du \]

Step 6: Solve each part: 1. First integral: \[ \int \frac{2u}{2 + u^2} du = \ln|2 + u^2| + C_1 \] 2. Second integral: \[ \int \frac{3}{2 + u^2} du = \frac{3}{\sqrt{2}} \tan^{-1}\left(\frac{u}{\sqrt{2}}\right) + C_2 \]

Step 7: Combine results and substitute back \( u = \tan x \): \[ \ln|\tan^2 x + 2| + \frac{3}{\sqrt{2}} \tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right) + C \]

Conclusion: The correct answer is \(\boxed{B}\) \(\left( \frac{3}{\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right) + \ln|\tan^2 x + 2| + C \right)\).

Answer 2

\[ \int \frac{2 \tan x + 3}{\sin^2 x + 2 \cos^2 x} \, dx \]

Step 1: Simplify the denominator

First, simplify the denominator \( \sin^2 x + 2 \cos^2 x \). Using the identity \( \sin^2 x + \cos^2 x = 1 \), we get:

\[ \sin^2 x + 2 \cos^2 x = 1 + \cos^2 x \]

Thus, the integral becomes:

\[ \int \frac{2 \tan x + 3}{1 + \cos^2 x} \, dx \]

Step 2: Substitute and simplify the terms

We now focus on simplifying the integral further. Note that the numerator involves \( \tan x \), which is \( \frac{\sin x}{\cos x} \), and we also have \( \cos^2 x \) in the denominator. To tackle this, we will use a substitution that simplifies the trigonometric terms.

Let's make the substitution \( t = \tan x \), so that \( dt = \sec^2 x \, dx \), and we know that \( \sin^2 x = \frac{t^2}{1 + t^2} \).

After applying this substitution and simplifying the integral, the result will lead us to the correct answer.

The correct form of the integral after simplification leads to an expression that matches option (B):

\[ \frac{3}{\sqrt{2}} \tan^{-1} \left( \frac{\tan x}{\sqrt{2}} \right) + \ln | \tan^2 x + 2 | + C \]