Chapter 7 Integrals | Exercise 7.5 I NCERT Solutions I New NCERT solution Class 12 I Class 12 Maths

Chapter 7 Integrals | Exercise 7.5 I NCERT Solutions I New NCERT solution Class 12 I Class 12 Maths

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[jeeva]

sir you're better than 90% of maths teachers in india.
so grateful to learn from you🙌🙌

[___anvesha]

Thankyou so much sir ! I bet aapke jaisa koi nhi smjha skta. Great Explaination and detailed 💛

[vatsavbitra]

such flawless and super explnation i love maths becasue of you sir it has made it so easy for me to learn and not feel anixety while doing maths.

[hashimzaid]

Best teacher.... everything understood very clearly 👍

[DefinitelyPrateek]

0:18 Imp Question
8:11 Imp qsn (t ka coefficient divide me ayega)
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[tanyasingh]

Thankyou so much sir, your explanation from basics are the best. You made integration so easy for us. 🙏

[akashkumarsingh]

Sir aapse math padhne ke baad math ek dam aasan ho jata hai mere liye ❤❤❤

[studyplace]

sir apne jasie 10th ka content bnaya h plz vaisa 12th ke liye bna dejiye

[Shloktiwadiii]

Thank you so much sir mujhe 12th ke start meh lga tha ki kese kru ga maths but now apki vajah seh sab aasan lag rha h ❤

[TheKrish-eqwt]

Sure! Here's something fun about integration and specifically about the technique of **Integration by Partial Fractions**:

### Fun Fact: Integration and the Quest for Simplicity

**Integration** is often seen as the art of "undoing" differentiation, which can lead to some surprisingly elegant solutions to complex problems. One of the reasons integration is so fascinating is that it often reveals a deeper simplicity within seemingly complicated expressions.

### The Fun in Integration by Partial Fractions

**Integration by Partial Fractions** is like breaking down a complicated puzzle into smaller, more manageable pieces. Imagine you’re given a challenging integral that looks like a big, messy fraction. Instead of tackling it head-on, you break it down into simpler fractions that are much easier to integrate individually.

#### A Fun Analogy:
Think of Integration by Partial Fractions like trying to split a large, complex cake (your complicated integral) into smaller slices that you can eat one by one.

1. **The Cake (Complex Fraction):** You start with a big cake, maybe with multiple layers and flavors mixed together (your complex fraction). It looks difficult to consume as a whole.

2. **Slicing the Cake (Partial Fractions):** You slice the cake into neat, simpler pieces (partial fractions), making it easier to enjoy each flavor individually.

3. **Eating the Cake (Integrating Each Fraction):** Now, you can savor each slice, appreciating its simplicity, and add up all the flavors to get back to the full experience (sum of the integrated partial fractions).

### Historical Twist:
Partial fractions were first systematically used by the Swiss mathematician Johann Bernoulli in the 17th century. Interestingly, the technique itself is rooted in the idea of reverse engineering—the idea that any complex algebraic fraction can be expressed as the sum of simpler ones, a concept that connects deeply to how mathematicians and engineers think about problem-solving in general: break down the problem, solve the simple parts, and then combine them.

### Example:
If you have an integral like \(\int \frac{1}{(x-1)(x+2)} dx\), you could break it down into:

\[
\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}
\]

Where \(A\) and \(B\) are constants that you solve for. Once you find those constants, the integral becomes much simpler and can be solved easily. It’s almost like solving a puzzle where you figure out the missing pieces, making what seemed complicated suddenly straightforward and enjoyable!

So, integration by partial fractions is not just a powerful mathematical tool—it’s a process that can transform a daunting challenge into a fun and satisfying experience, much like savoring each slice of a delicious cake!

[unknownmee]

You are God of teaching maths for me!!!

[Lakshyarts-dl]

awesome session sir all concepts are cleared😇😇😇😇😇😇😇😇😇😇😇😇😇 bas aise hi video dalte raho 🙂 board exam me phod denge

[shiveshs_]

Sir where is secret folder 📁??
It helped a lot during boards exams ( 10th). I scored 100 because of it. Sir please give us that folder

[bobbychaudhary]

sir 7.5 mai question no.15 mai direct bhe toh kr sakte hai identity banake pr answer thoda alag arra hai ncert se .

[anirudhtiwari]

blessed because I got teacher like you sir💫. Salute to your hardwork sir🙌

[Lakshyarts-dl]

kon kon board exam me fodna💥💥💥 chahta hai ek baar josh💫😤 ke saath yes likhdo🎖🎖🎖🙋‍♀🙋‍♀🙋‍♀

[Cricketers.]

Very nice vdo sir so much....sab smjh agya ❤

[GenZ-Gamer]

Thnk u so much sir❤❤❤u made Integral to easy and too simple 😊❤❤❤❤again thank u so much

[omkarjha]

Sir aapse ek request hai. Please sir thumbnail mai ye bhi add kr dijiyega ki question no kitna se kitna h

[omkarjha]

Awesome lecture . Your teaching style is amazing sir ❤❤❤❤❤❤