INTEGRALS in 1 Shot- All Concepts with PYQs | Class 12 NCERT

Introduction

This tutorial covers the concepts of integration as presented in the video "INTEGRALS in 1 Shot" by NCERT Wallah. It aims to provide a comprehensive understanding of various integration techniques that are essential for Class 12 mathematics. By following this guide, students will gain clarity on the methods of integration, their properties, and applications in solving problems.

Step 1: Understand Properties of Indefinite Integration

Familiarize yourself with the fundamental properties of indefinite integration, which serve as the basis for solving integration problems.

• Key Properties:
  • Linearity:
    • ∫(a * f(x) + b * g(x)) dx = a * ∫f(x) dx + b * ∫g(x) dx
  • Constant of Integration:
    • Always add "+ C" after integrating.

Step 2: Learn Basic Formulas

Memorize essential integration formulas to apply them effectively in problems.

• Common Formulas:
  • ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1
  • ∫e^x dx = e^x + C
  • ∫sin(x) dx = -cos(x) + C
  • ∫cos(x) dx = sin(x) + C

Step 3: Master Integration by Substitution

This technique simplifies complex integrals by substituting a part of the function with a new variable.

• Steps for Substitution:
  1. Identify a suitable substitution, u = g(x).
  2. Compute du = g'(x) dx.
  3. Rewrite the integral in terms of u.
  4. Integrate with respect to u.
  5. Substitute back to x.

Practical Tip:

Choose a substitution that simplifies the integral significantly.

Step 4: Apply Integration Using Trigonometric Identities

Use trigonometric identities to simplify integrals involving trigonometric functions.

• Example:
  • To integrate ∫sin^2(x) dx, use the identity sin^2(x) = (1 - cos(2x))/2.

Step 5: Explore Integration by Partial Fractions

This method is useful for integrating rational functions.

• Steps for Partial Fractions:
  1. Factor the denominator.
  2. Express the rational function as a sum of simpler fractions.
  3. Integrate each fraction separately.

Common Pitfall:

Ensure that the degree of the numerator is less than the degree of the denominator before applying this method.

Step 6: Learn Integration by Parts

Utilize this technique when the integral is a product of two functions.

• Formula:
  • ∫u dv = uv - ∫v du
  • Choose u and dv strategically to simplify the integral.

Step 7: Understand Definite Integration

Definite integrals calculate the area under the curve between two limits.

• Steps:
  1. Compute the indefinite integral.
  2. Apply the limits by substituting them into the antiderivative.
  3. Subtract the results.

Step 8: Use Substitution in Definite Integration

Apply the substitution method in definite integrals for simpler computations.

• Steps:
  1. Perform the substitution as usual, adjusting the limits according to the new variable.
  2. Calculate the integral with the new limits.

Step 9: Review Properties of Definite Integration

Understand the key properties that govern definite integrals.

• Important Properties:
  • ∫[a to b] f(x) dx = -∫[b to a] f(x) dx
  • ∫[a to a] f(x) dx = 0
  • ∫[a to b] (f(x) + g(x)) dx = ∫[a to b] f(x) dx + ∫[a to b] g(x) dx

Conclusion

This tutorial has outlined the essential concepts and techniques of integration, including substitution, partial fractions, and integration by parts. By mastering these steps, you will enhance your problem-solving skills in calculus. Practice these methods with various problems to solidify your understanding and prepare for exams. For further learning, consider exploring more advanced topics in calculus or reviewing previous year questions based on integration.