818 marks

Integral calculus

This chapter deals with integration, both indefinite and definite, and its applications in finding areas and solving differential equations.

12 Topics

55h prep

5.6% subject weight

12 Topics

Integral as an anti-derivative

2m2/10

📌 Key Formula∫ f(x) dx = F(x) + C, where F'(x) = f(x)

Fundamental Integrals involving algebraic and trigonometric functions

2m2/10

📌 Key Formula∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

Fundamental Integrals involving exponential and logarithms functions

2m2/10

📌 Key Formula∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C, ∫ 1/x dx = ln|x| + C

Integrations by substitution

2m3/10

📌 Key Formula∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)

Integration using trigonometric identities

2m3/10

📌 Key FormulaUsing identities like sin²x = (1-cos2x)/2, cos²x = (1+cos2x)/2, etc.

Evaluation of simple integrals of the type ∫ dx/x^2+a^2 , ∫ dx √x^2 ± a2 , ∫ dx / a^2− x^2 , ∫dx/ √a^2− x^2 , ∫ dx/ ax^2+bx+c ,∫ dx/ √ax^2+ bx+c , ∫ (x+q)dx / ax^2+bx+c , ∫ (px+q)dx/ √ax^2+ bx+c , ∫ √a^2 ± x^2 dx , ∫√x^2 − a^2 dx

Evaluation of simple integrals of the type ∫ dx/x^2+a^2 , ∫ dx √x^2 ± a2 , ∫ dx / a^2− x^2 , ∫dx/ √a^2− x^2 , ∫ dx/ ax^2+bx+c ,∫ dx/ √ax^2+ bx+c , ∫ (x+q)dx / ax^2+bx+c , ∫ (px+q)dx/ √ax^2+ bx+c , ∫ √a^2 ± x^2 dx , ∫√x^2 − a^2 dx

2m4/10

📌 Key FormulaStandard formulas for these types of integrals.

Integral as limit of a sum

2m5/10

📌 Key Formula∫ₐᵇ f(x) dx = lim_{n→∞} Σ f(a + kΔx) Δx, where Δx = (b-a)/n

The fundamental theorem of calculus

2m4/10

📌 Key Formulad/dx ∫ₐˣ f(t) dt = f(x), ∫ₐᵇ f(x) dx = F(b) - F(a), where F'(x)=f(x)

Properties of definite integrals

2m4/10

📌 Key Formula∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx, ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx, ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx

Evaluation of definite integrals

2m4/10

📌 Key FormulaUsing substitution, properties, and fundamental theorem.

Determining areas of the regions bounded by simple curves in standard form

2m5/10

📌 Key FormulaArea between curve y = f(x) and x-axis: ∫ₐᵇ |f(x)| dx. Area between two curves: ∫ₐᵇ |f(x) - g(x)| dx

Integrations by partial functions

2m3/10

📌 Key Formula∫ P(x)/Q(x) dx after splitting into partial fractions.

818 marks

Integral calculus

This chapter deals with integration, both indefinite and definite, and its applications in finding areas and solving differential equations.

12 Topics

55h prep

5.6% subject weight

12 Topics

Integral as an anti-derivative

2m2/10

📌 Key Formula∫ f(x) dx = F(x) + C, where F'(x) = f(x)

Fundamental Integrals involving algebraic and trigonometric functions

2m2/10

📌 Key Formula∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

Fundamental Integrals involving exponential and logarithms functions

2m2/10

📌 Key Formula∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C, ∫ 1/x dx = ln|x| + C

Integrations by substitution

2m3/10

📌 Key Formula∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)

Integration using trigonometric identities

2m3/10

📌 Key FormulaUsing identities like sin²x = (1-cos2x)/2, cos²x = (1+cos2x)/2, etc.

Evaluation of simple integrals of the type ∫ dx/x^2+a^2 , ∫ dx √x^2 ± a2 , ∫ dx / a^2− x^2 , ∫dx/ √a^2− x^2 , ∫ dx/ ax^2+bx+c ,∫ dx/ √ax^2+ bx+c , ∫ (x+q)dx / ax^2+bx+c , ∫ (px+q)dx/ √ax^2+ bx+c , ∫ √a^2 ± x^2 dx , ∫√x^2 − a^2 dx

2m4/10

📌 Key FormulaStandard formulas for these types of integrals.

Integral as limit of a sum

2m5/10

📌 Key Formula∫ₐᵇ f(x) dx = lim_{n→∞} Σ f(a + kΔx) Δx, where Δx = (b-a)/n

The fundamental theorem of calculus

2m4/10

📌 Key Formulad/dx ∫ₐˣ f(t) dt = f(x), ∫ₐᵇ f(x) dx = F(b) - F(a), where F'(x)=f(x)

Properties of definite integrals

2m4/10

📌 Key Formula∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx, ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx, ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx

Evaluation of definite integrals

2m4/10

📌 Key FormulaUsing substitution, properties, and fundamental theorem.

Determining areas of the regions bounded by simple curves in standard form

2m5/10

📌 Key FormulaArea between curve y = f(x) and x-axis: ∫ₐᵇ |f(x)| dx. Area between two curves: ∫ₐᵇ |f(x) - g(x)| dx

Integrations by partial functions

2m3/10

📌 Key Formula∫ P(x)/Q(x) dx after splitting into partial fractions.