Integration by Parts ka Solution - Class 12 Maths

Integration by parts is your go-to tool for cracking those gnarly integrals that just won’t budge with basic formulas. Honestly, it’s like the reverse gear of the product rule in differentiation. You’ve got a product of two functions staring at you—ugly, messy—and instead of panicking, you flip the script. That’s the trick. It breaks the thing down, swaps one part for its derivative and the other for its integral, and suddenly the problem feels lighter. For Class 12, it’s a game-changer, trust me.

Isme Kya Cover Hai?

• Isme Kya Cover Hai? [PARA] Integration by parts ka basic formula—woh classic equation jo aapko har jagah milegi, usse hum thoda tod kar dekhenge. Pehle uski asal shape samjho, phir uske peechhe ka logic khud nikalna shuru karo.
• Real-life uses? Let’s get into it, because that’s where this thing actually proves its worth. You’re not just buying a feature list here—you’re buying something that works when you need it most. Think day-to-day tasks that pop up out of nowhere, and you need a quick, solid answer. That’s where it shines. It handles the routine stuff, the messy scenarios you didn’t plan for, and everything in between. Honestly, once you start using it, you wonder how you managed without it. It’s that practical. Real situations, real fixes—that’s the heart of it. No fluff.
• Straight up, this thing is packed with step-by-step solved examples taken directly from the NCERT textbooks. So you're not just getting random problems. You're getting the exact ones from your syllabus, all worked out in a way that's super easy to follow along with. Each step is laid out clearly. You never get lost in the middle of a solution.

Simple — yeh poora content CBSE syllabus ke hisaab se hi banaya gaya hai. Koi guess nahi, koi random topics nahi. Jo padhna hai, wahi milega.

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 mein ek aisa hathyar hai, jisse hum do functions ke product ka integral nikal sakte hain. Aur suno, iska istemal aksar tab hota hai jab substitution method fail ho jaye — bilkul wahi moment jab lagta hai ki ab kya karein, tabhi ye technique kaam aati hai.

Integration by Parts ka Formula

The formula is straightforward: ∫u dv = uv - ∫v du. You’ve got two functions here, u and v. To actually get this, you take the product rule and integrate it. That’s the whole trick behind it.

U aur V kaise Choose Karen?

So, aap U choose karna seekh rahe hain? Simple hai — bas LIATE rule yaad rakhiye. Ye rule batata hai ki kis order me U select karna chahiye: Logarithmic, phir Inverse trigonometric, phir Algebraic, phir Trigonometric, aur last me Exponential. Matlab, jo pehle aata hai list me, usko U bana lijiye. Samjhiye ek example se. Agar integral me ln x dikh raha hai, toh wahan pe U = ln x lena sabse behtar rahega. Aur haan, ye trick kaafi kaam aati hai jab aap complex integrals se jhoojh rahe hote hain.

Step-by-Step Solution Process

1. Pehle integral ko dekhte hi, u aur dv ko identify karna zaroori hai. Bina soche-samjhe aage badhna galat hoga—pehle in dono ko sahi se chuno.
2. Step one, you differentiate — then you integrate. Simple as that—just run the math both ways to nail down the du and v.
3. Here’s the thing—this is the part where you actually plug your values into the formula ∫u dv = uv − ∫v du. Just swap in whatever you've got for u, dv, and the rest. Straight substitution. No magic, no shortcuts. You're basically taking the pieces you picked earlier and dropping them right into place. That’s the whole move right there.
4. Last, solve the resulting integral and don’t forget to slap on that constant of integration.

Solved Example 1: ∫x e^x dx

Here’s the thing — integration by parts is our tool here, and it works like a charm. So we start by picking u = x and dv = e^x dx. That gives us du = dx, and v? Well, that’s just e^x again, nice and clean. Now we plug into the formula: ∫x e^x dx becomes x e^x minus ∫e^x dx. And since ∫e^x dx is e^x, we’re left with x e^x - e^x, plus our constant C. Tidy it up a bit. Factor out the e^x — and you get e^x (x - 1) + C. Done. That’s your answer, plain and simple.

Solved Example 2: ∫x sin x dx

Here’s the solution. Start by picking u = x and dv = sin x dx. That gives you du = dx and v = -cos x. Now plug everything into the integration by parts formula. So you get ∫x sin x dx = x(-cos x) - ∫(-cos x) dx. Clean that up a bit: -x cos x + ∫cos x dx, and the integral of cos x is just sin x. So the final answer is -x cos x + sin x + C. Easy enough once you see the pattern.

Common Mistakes aur Avoid Karne ke Tips

• U aur dv ko galat chunna—ho gaya na integral ka complex banna. Sirf ek chhota sa galti, aur poora problem ulajh jata hai. Isliye hamesha dhyan se socho ki kaunsa part differentiate karna hai aur kaunsa integrate.
• Integration constant bhoolna? Haan, woh bilkul aisa hi hai jaise exam ke din calculator ghar chhod dena—poore paper ka nuksan. Teachers isko dekh ke seedha marks kaat dete hain, koi mercy nahi. Ek chhota sa “+C” likhna bhool gaye, aur aapka poora answer galat ho jata hai. Isliye, har baar integration karte waqt, aakhri step pe ruk kar check karo—woh constant wahan hona chahiye. Aur haan, practice karo, kyunki yeh galti sirf ek baar hoti hai jab aap isse seriously lete ho.
• Look, these are the kinds of mistakes that sneak up on you—simple algebraic errors during substitution or simplification. You know, the classic mix-up with signs, or maybe dropping a term when you're in a rush. The annoying part is that they’re so easy to make, yet they wreck the whole answer. During substitution, if you’re not careful about where the parentheses go, boom, everything falls apart. And simplification? That’s where people get lazy and start skipping steps. Big mistake. You think you’re saving time, but you’re just inviting trouble. The trick is to slow down, write out every little step, and double-check your work. Because these errors are avoidable, plain and simple.
• Honestly, the biggest mistake people make? Ignoring the LIATE rule. You think you're saving time by skipping it, but really, you're just setting yourself up for a mess. Ten minutes later, you're stuck in a loop, going in circles, and wondering why you didn't just follow the damn thing. It's a shortcut that costs you way more than it saves. Trust the rule — it's there for a reason.

NCERT Exercises se Related Examples

NCERT ke Chapter 7, Integrals mein jo examples diye gaye hain, unmein se ek hai ∫x^2 e^x dx. Isse integration by parts se solve karte hain. Chalte hain step-by-step—hum u = x^2 lete hain, aur dv = e^x dx. Phir bas formula apply kar dete hain. Itna hi hai.

Additional Practice Problems

A few more to grind through before the big day: ∫e^x cos x dx, ∫x ln x dx, ∫arctan x dx. Just work them out—one by one. You'll feel way more solid by then, no doubt.

Conclusion

Integration by parts samajh liya? Toh aadhe tricky integrals basically solve ho jaate hain—almost like magic. Thoda regular practice, aur NCERT ke examples ko seriously lo—skip mat karna, kyunki wahi base banate hain. Honestly, yeh formula pehle darata hai, but once it clicks? You're golden. Practice hi asli key hai, aur examples ko ignore karna? That's a rookie mistake. Toh haan, consistent effort se, aap confidently exam mein accha score kar sakte hain—bas itna hi.