Integration: Class 12 Mathematics

Key points

Integration is the reverse of differentiation. If d/dx F(x) = f(x), then F(x) is an anti-derivative of f(x), and we write ∫ f(x) dx = F(x) + C. This is the indefinite integral. C is the constant of integration, because the derivative of any constant is zero.

Basic rules:

• ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1.
• ∫ 1/x dx = ln |x| + C.
• ∫ k f(x) dx = k ∫ f(x) dx, where k is a constant.
• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

Standard integrals (x in radians):

• ∫ eˣ dx = eˣ + C and ∫ aˣ dx = aˣ/ln a + C.
• ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C.
• ∫ sec² x dx = tan x + C and ∫ cosec² x dx = −cot x + C.
• ∫ sec x tan x dx = sec x + C.
• ∫ 1/(1 + x²) dx = tan⁻¹x + C.

Integration of a linear function inside: if ∫ f(x) dx = F(x) + C, then ∫ f(ax + b) dx = F(ax + b)/a + C. For example, ∫ cos(2x + 1) dx = sin(2x + 1)/2 + C.

Integration by substitution: choose a part of the integrand as u so that du appears too. Replace everything in terms of u, integrate, and then return to x. A very common form is ∫ f′(x)/f(x) dx = ln |f(x)| + C. For example, ∫ 2x/(x² + 1) dx = ln(x² + 1) + C, since the top is the derivative of the bottom.

Useful trigonometric identities for integration are sin² x = (1 − cos 2x)/2 and cos² x = (1 + cos 2x)/2. So ∫ sin² x dx = x/2 − (sin 2x)/4 + C.

Integration by parts is used for a product of two different kinds of function. The formula is ∫ u dv = uv − ∫ v du. Choose u in the order of ILATE: inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. For ∫ x eˣ dx take u = x and dv = eˣ dx, so the answer is x eˣ − eˣ + C.

Partial fractions are used to integrate a rational function whose denominator can be factorised. For example, 1/[(x − 1)(x + 2)] = (1/3)[1/(x − 1) − 1/(x + 2)], so its integral is (1/3) ln |(x − 1)/(x + 2)| + C.

The definite integral from a to b is ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any anti-derivative. There is no constant C because it cancels. This is the fundamental theorem of calculus.

Properties of definite integrals:

• ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx.
• ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx for a point c between a and b.
• ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx.
• ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even, and 0 if f is odd.

Area under a curve: the area between y = f(x), the x-axis, and the lines x = a and x = b is ∫ₐᵇ f(x) dx when f(x) ≥ 0 on [a, b]. If the curve lies below the x-axis, the integral is negative, so take its absolute value for the area. The area between two curves is ∫ₐᵇ [upper − lower] dx. When substituting in a definite integral, change the limits to match the new variable.

Study tip: Learn the standard integrals as a table, and always differentiate your answer to check it, because this takes only a few seconds. For substitution, write the choice of u and du clearly, since marks are given for the method. In area questions, sketch the curve first, mark the limits, and give the answer in square units.

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Common mistakes

• Forgetting the constant C in an indefinite integral, and wrongly adding C in a definite integral.
• Using the power rule for ∫ 1/x dx, which gives division by zero. The answer is ln |x| + C.
• Not dividing by the coefficient of x, for example writing ∫ e²ˣ dx = e²ˣ instead of e²ˣ/2.
• Forgetting to change the limits when substituting in a definite integral, or to return to x in an indefinite one.
• Taking the area as a negative number. Area is never negative, so use the absolute value for the part below the axis.

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