Cosec x ka integration kya hoga Hindi me solve kar do.

Cosec \( x \) ka integration karne ke liye, hum standard integration formula ka use karte hain.

### Problem: \(\int \csc x \, dx\) 

**Step 1:** Hum \(\csc x\) ko \( \frac{1}{\sin x} \) likh sakte hain. Ab hum isko solve karne ke liye ek trick ka use karte hain. Hum numerator aur denominator dono ko \(\csc x + \cot x\) se multiply aur divide karenge.

\[
\int \csc x \, dx = \int \frac{\csc x}{1} \, dx = \int \frac{\csc x (\csc x + \cot x)}{(\csc x + \cot x)} \, dx 
\]

**Step 2:** Numerator ko expand karte hain:

\[
\int \frac{\csc^2 x + \csc x \cot x}{\csc x + \cot x} \, dx 
\]

Ab, hum substitution karenge. Let:

\[
u = \csc x + \cot x
\]

Toh, \( du \) kya hoga?

\[
du = -(\csc x \cot x + \csc^2 x) \, dx 
\]

Isko rewrite karte hain:

\[
du = -(\csc^2 x + \csc x \cot x) \, dx
\]

Isko simplify karne par:

\[
\int \csc x \, dx = -\int \frac{du}{u}
\]

**Step 3:** Ab, standard integration formula \( \int \frac{1}{u} \, du = \ln |u| + C \) ka use karte hain:

\[
\int \csc x \, dx = -\ln |\csc x + \cot x| + C
\]

Yeh final answer hai.

### Answer:

\[
\int \csc x \, dx = -\ln |\csc x + \cot x| + C
\]

**Hindi Explanation:**

- Pehle, humne \(\csc x\) ko \( \frac{1}{\sin x} \) likha.
- Fir, numerator aur denominator ko \( \csc x + \cot x \) se multiply aur divide kiya.
- Phir substitution method ka use kiya jisme \( u = \csc x + \cot x \) let kiya aur differentiate kiya.
- Iske baad, integration formula \( \int \frac{1}{u} du = \ln |u| + C \) ka use kiya.
- Ant mein, substitution ko wapas replace karke final answer mila.

Kya aapko koi aur details chahiye ya koi questions hain? 

Here are five questions related to this topic:
1. Kya aapko substitution method ke baare mein aur jaanana hai?
2. Kya aap aur trigonometric functions ke integrals seekhna chahenge?
3. Kya aapko trigonometric identities ke baare mein aur examples chahiye?
4. Kya aapko definite integrals ke baare mein bhi seekhna hai?
5. Kya aap integration by parts method ke baare mein aur seekhna chahenge?

**Tip:** Substitution method trigonometric integrals ke liye bahut effective hota hai. Hamesha pehle ye check karein ki koi appropriate substitution ho sakta hai ya nahi.

Learn how to integrate cosecant function (csc x) step-by-step in Hindi. Understand the use of substitution method and the integration formula ∫(csc x dx) = -ln|csc x + cot x| + C. Ideal for college-level students and those preparing for competitive exams.

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