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Integration Questions with Solutions for Class 11 and 12 | Testbook
IMPORTANT LINKS
We have put together a list of integration questions and answers that are perfect for Class 11 and Class 12 students. Integration is an important topic in higher-level math, and learning it well now will help you in future studies. These questions follow the CBSE board and NCERT syllabus.
By practicing these problems, you will gain confidence and improve your ability to solve difficult questions. This will help you score better in exams.
Maths Notes Free PDFs
| Topic | PDF Link |
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| Class 12 Maths Important Topics Free Notes PDF | Download PDF |
| Class 10, 11 Mathematics Study Notes | Download PDF |
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∫1 dx = x + C
∫a dx = ax + C
∫xⁿ dx = (xⁿ⁺¹ / (n + 1)) + C ; n ≠ -1
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec² x dx = tan x + C
∫csc² x dx = -cot x + C
∫sec x · tan x dx = sec x + C
∫csc x · cot x dx = -csc x + C
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫aˣ dx = (aˣ / ln a) + C ; a > 0, a ≠ 1
For additional formulas, you can visit this page: Integration
Integration Questions with Step-by-Step Solutions
Let's explore some questions on integration along with their detailed solutions.
- Find ∫(3x² + 2x + 1) dx
Solution:
Break into parts:
∫3x² dx + ∫2x dx + ∫1 dx
= 3 ∫x² dx + 2 ∫x dx + ∫1 dx
= 3(x³/3) + 2(x²/2) + x + C
= x³ + x² + x + C
- Compute ∫2x cos (x 2 – 3) dx.
Solution: Let, I = ∫2xcos(x 2 – 3) dx.
Now, let x 2 – 3 = u …..(1)
Then, 2x.dx = du.
Substituting these values, we get
I = ∫cos(u) du
= sin u + C …..(2)
Substituting the value from equation 1 in 2, we get
= sin (x 2 – 3) + C.
- Find ∫x² dx
Solution:
We use the formula: ∫xⁿ dx = (xⁿ⁺¹ / (n + 1)) + C
Here, n = 2
So,
∫x² dx = x³ / 3 + C
- Find ∫e^x dx
Solution:
We use the formula: ∫eˣ dx = eˣ + C
So,
∫e^x dx = e^x + C
- Find ∫1/x dx
Solution:
We use the formula: ∫1/x dx = ln|x| + C
So,
∫1/x dx = ln|x| + C
Practice Questions
- ∫x³ dx
- ∫cos x dx
- ∫(2x² + 5x + 3) dx
- ∫1/x dx
- ∫e^x dx
- ∫x⁴ + 2x² + 1 dx
- ∫sec²x dx
More Resources
FAQ’s For Integration
What is integration in maths?
Integration is a fundamental concept in calculus that combines functions together to form a new function. It's often used to find areas, volumes, central points, and many useful things.
What are the common integral formulas used to solve integration problems?
Common integral formulas include ∫1 dx=x+C, ∫a dx=ax+C, ∫x^n dx=(x^(n+1))/(n+1)+C for n ≠-1, ∫sin x dx=-cos x+C, and more.
What are the basic types of integration?
There are two main types: Definite Integration – gives a specific value. Indefinite Integration – gives a general formula including a constant of integration (C).
What is the symbol for integration?
The symbol for integration is ∫ (called an integral sign).
What is the difference between definite and indefinite integrals?
Definite integrals have limits and give a numerical value. Indefinite integrals don’t have limits and give a function with a constant (C).
What is the constant of integration?
It's a constant (C) added to the result of an indefinite integral because differentiating any constant gives 0.
Where is integration used in real life?
It is used in calculating areas, volumes, physics (motion), economics, and statistics.
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