Integration and Definite Integrals

(\int) is a mathematical symbol that represents the integral of a function. It is used to find the area under the curve of a function.

The Trapezoidal Rule is a basic numerical method for approximating definite integrals.

The equation (\int_0^1 \frac{1}{1+x^2} dx = \frac{\pi}{4}) is known as the Wallis integral, which was first discovered by John Wallis in the 17th century.

The value of the integral (\int_C \frac{1}{z} dz) is given by (2\pi i), where (C) is a positively oriented simple closed curve around the origin. This is known as Cauchy's integral theorem.

To evaluate the integral, we can use the power rule of integration. The integral of $x^n$ is $\frac{x^{n+1}}{n+1}$. Applying this rule, we get $\int_0^1 x^2 dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}$.

To evaluate the integral, we can use the power rule of integration: (\int_{0}^{1} x^2 dx = \left[\frac{x^3}{3}\right]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}).

To find the indefinite integral, we can use partial fraction decomposition to rewrite the integrand as (\frac{x^2 + 2x - 3}{x - 1} = x + 3 + \frac{1}{x - 1}). Then, we can integrate each term separately: (\int (x + 3 + \frac{1}{x - 1}) dx = \frac{x^2}{2} + 3x + \ln|x - 1| + C), where (C) is the constant of integration.

Using the power rule of integration, we get (\int_0^2 x^2 dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}).

The improper integral (\int_0^\infty \frac{1}{x} dx) diverges to (\infty). This is because the function (f(x) = \frac{1}{x}) has an infinite discontinuity at (x = 0), and the integral (\int_0^a \frac{1}{x} dx) diverges to (\infty) as (a) approaches (0).

Using the partial fraction decomposition, we can write (\frac{1}{x^2 - 4} = \frac{1}{2(x - 2)} + \frac{1}{2(x + 2)}). Integrating each term, we get (\int \frac{1}{x^2 - 4} dx = \frac{1}{2}\ln|x - 2| + \frac{1}{2}\ln|x + 2| + C), where (C) is the constant of integration.

The symbol ∫ is used to represent the integral of a function.

The integral of a function f(x) is given by the formula F(x) = ∫f(x) dx. Substituting the value of f(x), we get F(x) = ∫(2x^3 - 3x^2 + 4x - 5) dx. Integrating each term separately, we get F(x) = (2/4)x^4 - (3/3)x^3 + (4/2)x^2 - 5x + C. Simplifying this equation, we get F(x) = x^4 - x^3 + 2x^2 - 5x + C, where C is the constant of integration.

To find the indefinite integral of (f(x)), we apply the power rule of integration. The indefinite integral of (x^n) is (\frac{x^{n+1}}{n+1} + C), where (C) is the constant of integration.

To evaluate the definite integral, we first find the indefinite integral of the integrand (3x^2 - 2x + 1) using the power rule and the constant rule of integration. Then, we evaluate the indefinite integral at the upper and lower limits of integration and subtract the result at the lower limit from the result at the upper limit.

To solve this integral using integration by substitution, let (u = 3x). Then, (du = 3 dx). Substituting (u) and (du) into the integral, we get (\int \sin(3x) dx = \int \sin(u) \frac{1}{3} du). Now, we can integrate (\sin(u)) using the power rule of integration.