Key Concepts:
Indefinite Integral:
Represents a family of functions whose derivative is the given function. It includes an arbitrary constant \(C\): \[ \int f(x) \,dx = F(x) + C \]Definite Integral:
Represents the area under a curve between two limits \(a\) and \(b\): \[ \int_a^b f(x) \,dx = F(b) - F(a) \] where \( F(x) \) is the antiderivative of \( f(x) \).Basic Integration Rules:
\[ \int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1 \] \[ \int e^x \,dx = e^x + C \] \[ \int \sin x \,dx = -\cos x + C, \quad \int \cos x \,dx = \sin x + C \]Examples:
Example 1: Indefinite Integral
Evaluate: \[ \int (3x^2 + 4x - 5) \,dx \]Solution:
\[ \int (3x^2 + 4x - 5) \,dx = \frac{3x^3}{3} + \frac{4x^2}{2} - 5x + C \] \[ = x^3 + 2x^2 - 5x + C \] Thus, the answer is \( \boxed{x^3 + 2x^2 - 5x + C} \).Example 2: Definite Integral
Evaluate: \[ \int_1^3 (2x + 1) \,dx \]Solution:
Find the antiderivative: \[ \int (2x + 1) \,dx = x^2 + x \] Evaluate at the limits: \[ F(3) = 3^2 + 3 = 9 + 3 = 12, \quad F(1) = 1^2 + 1 = 1 + 1 = 2 \] \[ \int_1^3 (2x + 1) \,dx = 12 - 2 = 10 \] Thus, the answer is \( \boxed{10} \).Example 3: Finding Area Under a Curve
Find the area under \( f(x) = x^2 \) from \( x = 0 \) to \( x = 2 \).Solution:
\[ \int_0^2 x^2 \,dx = \frac{x^3}{3} \Big|_0^2 \] Evaluate at the limits: \[ \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} - 0 = \frac{8}{3} \] Thus, the area under the curve is \( \boxed{\frac{8}{3}} \).