Question 1

What is the difference between definite and indefinite integration?

Accepted Answer

Indefinite integration finds the general antiderivative of a function and includes a constant of integration (+c). It represents a family of functions. Definite integration evaluates the integral between two limits (a and b) and gives a numerical value representing the area under the curve from x=a to x=b. The Fundamental Theorem of Calculus links the two: the definite integral equals the difference of the antiderivative evaluated at the limits.

Question 2

How do I integrate by substitution?

Accepted Answer

Integration by substitution is used when the integrand contains a composite function. Choose a substitution u = g(x), then replace dx with du/g'(x). Simplify the integral in terms of u, integrate, then substitute back to x. For definite integrals, you can either change the limits to u-values or substitute back before evaluating. Common examples include ∫2x e^(x²) dx (let u = x²) and ∫sin x cos x dx (let u = sin x).

Question 3

When should I use integration by parts?

Accepted Answer

Use integration by parts when the integrand is a product of two functions that cannot be integrated by simple substitution. The formula is ∫u dv = uv - ∫v du. Choose u as the function that simplifies when differentiated (using LIATE: Log, Inverse trig, Algebraic, Trig, Exponential). For example, ∫x eˣ dx: let u = x, dv = eˣ dx. For ∫x² sin x dx, you may need to apply integration by parts twice.

Question 4

How do I find the area between two curves?

Accepted Answer

To find the area between two curves y = f(x) and y = g(x) from x=a to x=b, where f(x) ≥ g(x) on [a,b], integrate the difference: Area = ∫ₐᵇ [f(x) - g(x)] dx. First find the points of intersection by solving f(x) = g(x) to determine the limits a and b. If the curves cross, split the integral at the intersection point and take absolute values.

Question 5

What is the constant of integration and why is it important?

Accepted Answer

The constant of integration (+c) is added to the result of an indefinite integral because the derivative of a constant is zero. This means there are infinitely many antiderivatives that differ by a constant. For example, ∫2x dx = x² + c, since the derivative of x² + 5 is also 2x. In real-world problems, the constant is determined by an initial condition (e.g., given a point on the curve). Omitting +c in indefinite integrals will lose marks.

Question 6

How do I integrate trigonometric functions like sin²x or cos²x?

Accepted Answer

Integrate sin²x and cos²x using the double-angle identities: sin²x = (1 - cos2x)/2 and cos²x = (1 + cos2x)/2. Then integrate term by term. For example, ∫sin²x dx = ∫(1 - cos2x)/2 dx = x/2 - (sin2x)/4 + c. Similarly, ∫cos²x dx = x/2 + (sin2x)/4 + c. These identities are essential for integrating powers of trig functions.

H: Integration

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Related Topics in AQA vocational Mathematics

A: Proof

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series

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