Class 12 Mathematics Chapter 7: Integrals
CBSE 2026–27 | Detailed NCERT-aligned study resource
Integration is the inverse process of differentiation. The CBSE syllabus specifically includes substitution, partial fractions, integration by parts, standard forms, the Fundamental Theorem of Calculus and properties/evaluation of definite integrals. citeturn0search11
1. Indefinite Integral
If F′(x)=f(x), then ∫f(x)dx=F(x)+C. The constant C is essential because differentiation removes constants.
Integration: f(x) → F(x)+C
2. Substitution
Look for a composite expression together with its derivative. Example: ∫2x(x²+1)⁵dx. Put u=x²+1, du=2x dx. Therefore ∫u⁵du=u⁶/6+C=(x²+1)⁶/6+C.
3. Integration by Parts
Formula: ∫u dv=uv−∫v du. Choose u so that differentiation simplifies it.
Worked example: ∫xeˣdx. Let u=x, dv=eˣdx. Then du=dx, v=eˣ. Hence ∫xeˣdx=xeˣ−∫eˣdx=eˣ(x−1)+C.
4. Partial Fractions
For rational functions, factor the denominator and decompose into simpler fractions. Example: 1/(x²−1)=1/2[1/(x−1)−1/(x+1)]. Hence its integral is 1/2 ln|x−1|−1/2 ln|x+1|+C.
5. Standard Forms
| Form | Method |
|---|---|
| ∫dx/(x²−a²) | Partial fractions / logarithmic form |
| ∫dx/√(a²−x²) | sin⁻¹(x/a)+C |
| ∫dx/√(x²+a²) | Logarithmic standard form |
| ∫dx/(ax²+bx+c) | Complete square / standard form |
| ∫(px+q)/(ax²+bx+c)dx | Express numerator using derivative of denominator |
| ∫√(a²+x²)dx | Standard result / suitable substitution |
These families are explicitly part of the current CBSE syllabus. citeturn0search11
6. Definite Integrals
For an antiderivative F, the Fundamental Theorem gives ∫ₐᵇf(x)dx=F(b)−F(a), under the usual conditions.
7. Properties You Should Know
- ∫ₐᵃf(x)dx=0
- ∫ₐᵇf(x)dx=−∫ᵇₐf(x)dx
- ∫ₐᵇf=∫ₐᶜf+∫ᶜᵇf
- If f is odd, ∫₋ₐᵃf(x)dx=0.
- If f is even, ∫₋ₐᵃf(x)dx=2∫₀ᵃf(x)dx.
8. Worked Definite-Integral Example
Evaluate ∫₀¹2x(x²+1)²dx. Let u=x²+1. Limits change from 1 to 2. Integral=∫₁²u²du=[u³/3]₁²=7/3.
9. How to Choose a Method
- Try direct standard formula first.
- If a composite function and derivative appear together, try substitution.
- If the integrand is a product, consider integration by parts.
- If it is a rational expression, check partial fractions.
- For definite integrals, inspect symmetry and properties before lengthy calculation.
10. Common Errors
- Forgetting +C.
- Incorrect substitution limits.
- Dropping absolute-value signs in logarithmic answers.
- Using integration by parts when substitution is simpler.
11. Practice Set
- Evaluate ∫(3x²+2x)(x³+x²+1)⁴dx.
- Evaluate ∫x cos x dx by parts.
- Integrate 1/(x²−4) using partial fractions.
- Evaluate a definite integral using even/odd symmetry.
- Evaluate an integral of the form 1/√(a²−x²).
- Choose and justify an appropriate method for a mixed integral.
Common Mistakes — Chapter 7
- Forgetting +C in an indefinite integral.
- Using integration by parts when direct substitution is simpler.
- Forgetting to change limits after substitution in a definite integral.