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.

∫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

Let's explore some questions on integration along with their detailed solutions.

1. 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

2. 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.

3. Find ∫x² dx

Solution:

We use the formula: ∫xⁿ dx = (xⁿ⁺¹ / (n + 1)) + C
Here, n = 2

So,
∫x² dx = x³ / 3 + C

4. Find ∫e^x dx

Solution:

We use the formula: ∫eˣ dx = eˣ + C

So,
∫e^x dx = e^x + C

5. Find ∫1/x dx

Solution:

We use the formula: ∫1/x dx = ln|x| + C

So,
∫1/x dx = ln|x| + C

Practice Questions

1. ∫x³ dx
2. ∫cos x dx
3. ∫(2x² + 5x + 3) dx
4. ∫1/x dx
5. ∫e^x dx
6. ∫x⁴ + 2x² + 1 dx
7. ∫sec²x dx

More Resources

• Integration By Parts
• Integration By Substitution
• Methods Of Integration