Class 12 Mathematics ANTI-DERIVATIVES Notes

Class 12 Mathematics ANTI-DERIVATIVES Notes
Anti-Derivatives Notes | Nepal eNotes

Unit 16

Calculus

Class 12 Mathematics

Anti-Derivatives

Class 12 Mathematics – Anti-Derivatives Notes PDF

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1. Anti-Derivative & Indefinite Integral2. Basic Integrals 3. Integration by Substitution4. Integration by Parts 5. Definite Integrals6. Properties of Definite Integrals 7. Area Between Curves8. Worked Examples 9. Integration Strategy10. Common Mistakes 11. Important Exam Questions12. One-Minute Revision

NEB / CDC Focus

This unit treats integration as the reverse process of differentiation, applies standard/basic integrals, develops substitution and integration-by-parts methods, evaluates definite integrals, and uses definite integration to find area between curves.

1. Anti-Derivative and Indefinite Integral

Anti-Derivative

If F′(x)=f(x), then F(x) is an anti-derivative of f(x).

∫f(x)dx = F(x)+C.

The constant C is essential because every function F(x)+C has the same derivative f(x).

Differentiation and Integration Are Inverse Processes F(x)family: F(x)+C f(x)=F′(x) differentiate →← integrate
Figure 1: Integration reverses differentiation, up to an arbitrary constant.

2. Basic Integrals

FunctionIntegral
xⁿ, n≠−1∫xⁿdx=xⁿ⁺¹/(n+1)+C
1/x∫dx/x=ln|x|+C
eˣ∫eˣdx=eˣ+C
aˣ∫aˣdx=aˣ/ln a+C, a>0,a≠1
sin x∫sin x dx=−cos x+C
cos x∫cos x dx=sin x+C
sec²x∫sec²x dx=tan x+C
cosec²x∫cosec²x dx=−cot x+C
sec x tan x∫sec x tan x dx=sec x+C
cosec x cot x∫cosec x cot x dx=−cosec x+C
1/(1+x²)tan⁻¹x+C
1/√(1−x²)sin⁻¹x+C
Linearity
∫[af(x)+bg(x)]dx=a∫f(x)dx+b∫g(x)dx.
Example

Integrate 3x²−4x+5.

∫(3x²−4x+5)dx=x³−2x²+5x+C.

3. Integration by Substitution

Substitution reverses the chain rule. If u=g(x), then du=g′(x)dx.

∫f(g(x))g′(x)dx = ∫f(u)du.
  1. Choose an inner expression u.
  2. Find du.
  3. Rewrite the entire integral in u.
  4. Integrate.
  5. Substitute back in x.
Worked Example

Evaluate ∫2x(x²+1)⁵dx.

Let u=x²+1; then du=2x dx.

∫u⁵du=u⁶/6+C=(x²+1)⁶/6+C.

Substitution Reverses the Chain Rule Spot inner functionu=g(x) Convert differentialdu=g′(x)dx Integrate in uthen substitute back
Figure 2: A clean substitution changes both the expression and dx.

4. Integration by Parts

Integration by parts reverses the product rule.

∫u dv = uv−∫v du.
Choosing u

A common guide is LIATE: Logarithmic, Inverse-trigonometric, Algebraic, Trigonometric, Exponential. It is a heuristic, not a law.

Worked Example

Evaluate ∫x eˣ dx.

Take u=x, dv=eˣdx. Then du=dx, v=eˣ.

∫xeˣdx=xeˣ−∫eˣdx=eˣ(x−1)+C.

Worked Example

Evaluate ∫ln x dx.

Take u=ln x, dv=dx.

=xln x−∫1 dx=xln x−x+C.

Integration by Parts Choose u and dvdifferentiate u, integrate dv uv − ∫v dusimpler remaining integral
Figure 3: Choose u so that differentiating it makes the problem simpler.

5. Definite Integrals

Definite Integral

If F′(x)=f(x), then by the fundamental evaluation rule,

∫ab f(x)dx = F(b)−F(a).

No arbitrary constant is needed in the final value because it cancels between the endpoints.

Worked Example

Evaluate ∫₀² 3x² dx.

=[x³]₀²=8.

Definite Integral as Signed Area ab
Figure 4: A definite integral measures signed area relative to the x-axis.

6. Useful Properties of Definite Integrals

∫ₐᵃ f(x)dx=0 ∫ₐᵇ f(x)dx=−∫ᵦᵃ f(x)dx ∫ₐᵇ f(x)dx=∫ₐᶜ f(x)dx+∫𝚌ᵇ f(x)dx ∫₋ₐᵃ f(x)dx=0 if f is odd ∫₋ₐᵃ f(x)dx=2∫₀ᵃ f(x)dx if f is even

Symmetry can greatly shorten exam calculations, but verify whether the integrand is even or odd before using it.

7. Area Between Curves

For curves y=f(x) and y=g(x) with f(x)≥g(x) on [a,b],

Area=∫ₐᵇ [f(x)−g(x)]dx.

If the upper/lower curve changes, split the integral at intersection points.

Worked Example

Find the area between y=x and y=x² from their intersections.

Intersections: x=x² ⇒ x=0,1. On [0,1], x≥x².

Area=∫₀¹(x−x²)dx=[x²/2−x³/3]₀¹=1/6.

Area Between y=x and y=x² y=xy=x²
Figure 5: Area is the integral of upper curve minus lower curve.

8. More Worked Examples

Example 1

∫(2x+1)/(x²+x+4) dx. Let u=x²+x+4.

=ln(x²+x+4)+C.

Example 2

∫x cos x dx.

=x sin x+cos x+C.

Example 3

∫₀^π sin x dx=[−cos x]₀^π=2.

Example 4

∫₋₂² x³ dx=0 because the integrand is odd.

Example 5

∫₋₂² x² dx=2∫₀²x²dx=16/3 because x² is even.

Choosing an Integration Method Standard form?Basic integral Inner + derivative?Substitution Product?Parts Bounds?Definite
Figure 6: Recognizing the structure is often the hardest part of integration.

9. Integration Strategy

  1. Simplify algebraically before integrating.
  2. Compare the integrand with the standard integral table.
  3. Look for an inner expression and its derivative for substitution.
  4. For products such as algebraic×exponential or algebraic×trigonometric, test integration by parts.
  5. For definite integrals, integrate first and then apply the bounds.
  6. Use symmetry where valid.
  7. For area, solve intersections and identify the upper curve before setting up the integral.
  8. Differentiate your indefinite result to check it.

10. Common Mistakes

Missing +C

Indefinite integrals require the constant of integration.

Power rule at n=−1

∫x⁻¹dx=ln|x|+C, not division by zero.

Incomplete substitution

Rewrite the entire integrand and dx in the new variable.

Wrong parts sign

∫u dv=uv−∫v du.

Definite +C

Do not carry an arbitrary constant into a final definite-integral value.

Area sign

Area is non-negative; use upper minus lower and split intervals when required.

11. Formula Sheet

∫xⁿdx=xⁿ⁺¹/(n+1)+C, n≠−1∫dx/x=ln|x|+C ∫u dv=uv−∫v du∫ₐᵇf(x)dx=F(b)−F(a) Area=∫ₐᵇ(upper−lower)dx

12. Important Exam Questions

Short

  1. Define anti-derivative and indefinite integral.
  2. Explain why the constant of integration is needed.
  3. State integration by parts.
  4. State the definite-integral evaluation rule.

Long / Numerical

  1. Integrate using standard formulae.
  2. Integrate using substitution.
  3. Integrate using parts.
  4. Evaluate definite integrals, including symmetry cases.
  5. Find the area bounded by two curves.

13. One-Minute Revision

  • Integration reverses differentiation.
  • Indefinite integrals need +C.
  • Use substitution to reverse the chain rule.
  • Use parts to reverse the product rule.
  • ∫u dv=uv−∫v du.
  • Definite integral = upper anti-derivative value minus lower.
  • Odd functions integrate to 0 over symmetric limits.
  • Even functions can be doubled from 0 to a.
  • Area between curves is upper minus lower.

14. Diagram Practice

  • Derivative/integral inverse-process diagram.
  • Substitution flow.
  • Integration-by-parts flow.
  • Definite integral as signed area.
  • Area between two curves.
  • Integration method decision chart.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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Original Nepal eNotes page: Anti-Derivatives Notes.
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