Integration questions

Integration questions are an essential part of calculus and are widely used in Class 11, 12, and advanced mathematics. Just like differentiation helps us find rates and slopes, integration is the reverse process, helping us calculate areas, volumes, and accumulated quantities.

Whether you're preparing for CBSE, ICSE, JEE, NEET, or any competitive exam, understanding how to do integration and practicing integrals questions is crucial.

 

Table of Contents

• What is Integration?
• Define Integration with Examples
• Integral Formulas
• Step-by-Step Solved Integration Questions Using Basic Techniques
• Integrals Questions With Solutions
• Important Tips to Solve Integration Questions Fast
• Practice Integration Questions
• Conclusion
• FAQ on Integration Questions

 

What is Integration?

What is integration? In simple words, integration is the process of adding up small parts to find the whole. In mathematics, it refers to summing up infinitesimally small quantities to find areas under curves, total distance, or volume.

It is the inverse of differentiation. If
d/dx (F(x)) = f(x)

Then
∫ f(x) dx = F(x) + C
Where C is the constant of integration.

You must understand what is integration to solve problems in physics, mathematics, and engineering efficiently.

 

Define Integration with Examples

Let’s define integration properly:
Integration is a fundamental operation in calculus that combines infinitesimal data into a whole quantity. It helps determine the total accumulation of a quantity.

Example:

∫ x² dx = (x³)/3 + C

So whenever someone asks to define integration, say it's the process of finding an integral of a function which results in accumulation or area.

 

Integral Formulas 

Memorizing integral formulas is key to solving integration questions quickly.

Common Integral Formulas Table

Revise these integral formulas regularly to ace your integration questions.

 

Step-by-Step Solved Integration Questions Using Basic Techniques

1. Substitution Method

Question:
Solve ∫ 2x(x² + 1)³ dx

Step 1: Let u = x² + 1
Then, du/dx = 2x → du = 2x dx

Step 2: Replace in the integral:
∫ 2x(x² + 1)³ dx = ∫ u³ du

Step 3: Integrate:
∫ u³ du = u⁴ / 4 + C

Step 4: Substitute back u = x² + 1
Final Answer: (x² + 1)⁴ / 4 + C

 

2. Integration by Parts

Question:
Solve ∫ x·e^x dx

Step 1: Let
u = x → du = dx
dv = e^x dx → v = e^x

Step 2: Use integration by parts formula:
∫ u dv = uv − ∫ v du

Apply the formula:
= x·e^x − ∫ e^x dx
= x·e^x − e^x + C

Final Answer:
x·e^x − e^x + C

 

3. Partial Fractions

Question:
Solve ∫ 1/[(x + 1)(x + 2)] dx

Step 1: Express the integrand in partial fractions:
1/[(x + 1)(x + 2)] = A/(x + 1) + B/(x + 2)

Multiply both sides by (x + 1)(x + 2):
1 = A(x + 2) + B(x + 1)

Now solve for A and B:
Let x = −2 → 1 = A(0) + B(−1) → B = −1
Let x = −1 → 1 = A(1) + B(0) → A = 1

So:
∫ 1/[(x + 1)(x + 2)] dx = ∫ [1/(x + 1) − 1/(x + 2)] dx

Step 2: Integrate each term:
= ln|x + 1| − ln|x + 2| + C
= ln|(x + 1)/(x + 2)| + C

Final Answer:
ln|(x + 1)/(x + 2)| + C

 

4. Trigonometric Identities

Question:
Solve ∫ sin²x dx

Step 1: Use the identity:
sin²x = (1 − cos(2x)) / 2

Step 2: Rewrite the integral:
∫ sin²x dx = ∫ (1 − cos(2x))/2 dx
= (1/2) ∫ (1 − cos(2x)) dx

Step 3: Integrate:
= (1/2)[x − (1/2)sin(2x)] + C
= x/2 − sin(2x)/4 + C

Final Answer:
x/2 − sin(2x)/4 + C

 

Integrals Questions With Solutions

Q1. Solve: ∫ x² dx

Step 1: Use the power rule of integration:
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C

Step 2: Substitute n = 2
∫ x² dx = x³ / 3 + C

Final Answer:
x³ / 3 + C

 

Q2. Solve: ∫ 1/x dx

Step 1: Use the standard formula:
∫ 1/x dx = ln|x| + C

Final Answer:
ln|x| + C

 

Q3. Solve: ∫ e^x dx

Step 1: Apply the exponential rule:
∫ e^x dx = e^x + C

Final Answer:
e^x + C

 

Q4. Solve: ∫ cos(x) dx

Step 1: Use the trigonometric formula:
∫ cos(x) dx = sin(x) + C

Final Answer:
sin(x) + C

 

Q5. Solve: ∫ x·cos(x) dx

Step 1: Use integration by parts. Let:
u = x
dv = cos(x) dx
Then:
du = dx
v = sin(x)

Step 2: Apply the formula:
∫ u dv = u·v − ∫ v du
= x·sin(x) − ∫ sin(x) dx
= x·sin(x) + cos(x) + C

Final Answer:
x·sin(x) + cos(x) + C

 

Q6. Solve: ∫ 1 / (x² + 1) dx

Step 1: Use the standard formula:
∫ 1 / (x² + 1) dx = tan⁻¹(x) + C

Final Answer:
tan⁻¹(x) + C

 

Q7. Solve: ∫₀^π sin(x) dx

Step 1: Integrate sin(x):
∫ sin(x) dx = −cos(x)

Step 2: Apply the limits from 0 to π:
= [−cos(x)]₀^π
= −cos(π) + cos(0)
= −(−1) + 1 = 2

Final Answer: 2

 

Q8. Solve: ∫ ln(x) dx

Step 1: Use integration by parts. Let:
u = ln(x)
dv = dx
Then:
du = 1/x dx
v = x

Step 2: Apply the formula:
∫ u dv = u·v − ∫ v du
= x·ln(x) − ∫ x·(1/x) dx
= x·ln(x) − ∫ 1 dx
= x·ln(x) − x + C

Final Answer:
x·ln(x) − x + C

 

Q9.If a force F(x) = 4x acts on a body moving from x = 0 to x = 5, find the work done.

Formula:
Work = ∫ₐᵇ F(x) dx

Solution:
= ∫₀⁵ 4x dx
= 4 ∫₀⁵ x dx
= 4 [x² / 2]₀⁵
= 4 [(25 / 2) − 0] = 50

Final Answer:
Work = 50 units

 

Q10. Solve: ∫ x·√(x² + 1) dx

Step 1: Use substitution. Let:
u = x² + 1
Ten:
du = 2x dx → x dx = (1/2) du

Step 2: Rewrite the integral:
∫ x·√(x² + 1) dx = (1/2) ∫ √u du
= (1/2) ∫ u^(1/2) du

Step 3: Integrate:
= (1/2) · (2/3) u^(3/2)
= (1/3) u^(3/2)

Step 4: Substitute back u = x² + 1
= (1/3) · (x² + 1)^(3/2) + C

Final Answer:
(1/3)(x² + 1)^(3/2) + C

 

Important Tips to Solve Integration Questions Fast

• Learn and revise all integral formulas

• Identify if substitution, parts, or partial fraction is needed

• Always simplify the function before solving

• Don’t forget the constant C in indefinite integrals

• Practice 10+ integration questions daily

• Use a formula sheet for quick reference

 

Practice Integration Questions 

1. ∫ sin²x dx

2. ∫ (3x² + 2x + 5) dx

3. ∫ 1/(x² + 1) dx

4. ∫ x·√(x² + 1) dx

5. ∫ tan x dx

6. ∫ ln x dx

7. ∫ (e^x)/(1 + e^x) dx

8. ∫ 1/√(1 - x²) dx

9. ∫ sec x dx

10. ∫ 1/(x² - a²) dx

11. A car's velocity is given by v(t) = 2t. Find the distance travelled from t = 0 to t = 5.

12. Water is flowing into a tank at a rate of R(t) = 3t litres per hour. How much water flows into the tank from t = 0 to t = 4?
    
    

Conclusion

To conclude, integration questions are a core concept in calculus. By understanding what is integration, being able to define integration, and mastering all the integral formulas, you’ll become efficient in solving all types of integrals questions.

Practice different question types daily and apply what you've learned about how to do integration. The more you practice, the better your speed and confidence will become.

 

Related Links

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Differentiation Questions - Practice a wide range of differentiation problems, from finding rates of change to solving advanced calculus questions, to enhance your problem‑solving skills.

 

Frequently Asked Questions  on Integration Questions

1. What are the 5 basic integration formulas?

Here are the 5 basic integration formulas:

1. ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, where n ≠ -1

2. ∫ 1/x dx = ln|x| + C

3. ∫ e^x dx = e^x + C

4. ∫ a^x dx = a^x / ln(a) + C

5. ∫ sin(x) dx = -cos(x) + C

 

2. What is the integral of COSX?

The integral of cos(x) is:
∫ cos(x) dx = sin(x) + C

 

3. What is integration of 2x?

Apply the power rule:
∫ 2x dx = x² + C

 

4. How to integrate 3X?

Again, use the power rule:
∫ 3x dx = (3x²) / 2 + C

 

5. How to integrate e to 2x?

Use substitution:
∫ e^(2x) dx = (1/2)·e^(2x) + C

 

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