Integration by Parts ka Solution - Class 12 Maths

Integration by parts ek important method hai jo complex integrals solve karne me help karta hai. Yeh product rule ka reverse application hai.

Isme Kya Cover Hai?

• Integration by parts ka basic formula
• Uses aur real-life applications
• Step-by-step solved examples NCERT se

Yeh content CBSE syllabus ke according designed hai.

Links for Chapter-wise Download NCERT Solution for Class 12 गणितI in hindi Language

Here we have provided NCERT Solution for Class 12 गणितI in hindi Language, Just select the chapters below to get solution of the same:

प्रतिलोम त्रिकोणमितीय फलन

आव्यूह

सारणिक

सांतत्य तथा अवकलनीयता

अवकलज के अनुप्रयोग

Integration by Parts ka Parichay

Integration by parts calculus ka ek aisa technique hai jisse hum do functions ke product ka integral nikal sakte hain. Iska use often tab hota hai jab substitution method kaam nahi karta.

Integration by Parts ka Formula

Formula hai: ∫u dv = uv - ∫v du. Yahan u aur v do functions hain. Is formula ko derive karne ke liye product rule ka integration kiya jata hai.

U aur V kaise Choose Karen?

U choose karne ke liye LIATE rule follow karte hain: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Example ke liye, agar integral me ln x hai, toh u = ln x le sakte hain.

Step-by-Step Solution Process

1. Pehle integral me se u aur dv identify karen.
2. Phir du aur v calculate karen by differentiation aur integration.
3. Formula ∫u dv = uv - ∫v du me values substitute karen.
4. Last me resulting integral solve karen aur constant of integration add karen.

Solved Example 1: ∫x e^x dx

Solution: Let u = x aur dv = e^x dx. Toh du = dx aur v = e^x. Formula se: ∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C = e^x (x - 1) + C.

Solved Example 2: ∫x sin x dx

Solution: Let u = x aur dv = sin x dx. Toh du = dx aur v = -cos x. Formula se: ∫x sin x dx = x (-cos x) - ∫(-cos x) dx = -x cos x + ∫cos x dx = -x cos x + sin x + C.

Common Mistakes aur Avoid Karne ke Tips

• U aur dv galat choose karna, jisse integral complex ho jata hai.
• Integration constant bhool jana, jo exams me marks kat sakta hai.
• Simple algebraic errors during substitution aur simplification.
• LIATE rule ko ignore karna, jisse time waste hota hai.

NCERT Exercises se Related Examples

NCERT book me Chapter 7 Integrals me diye gaye examples jaise ∫x^2 e^x dx ko integration by parts se solve karna. Step-by-step, hum u = x^2 aur dv = e^x dx lete hain, phir formula apply karte hain.

Additional Practice Problems

Exam preparation ke liye kuch extra problems: ∫e^x cos x dx, ∫x ln x dx, ∫arctan x dx. Inko solve karke aap confidence build kar sakte hain.

Conclusion

Integration by parts samajhne ke baad, aap kai tricky integrals easily handle kar sakte hain. Regular practice aur NCERT examples se aap exams me accha perform kar sakte hain.