∫f(x) dx + ∫g(x) dx
∫[f(x) + g(x)] dx
The expression ∫f(x) dx + ∫g(x) dx represents the sum of two integrals. To evaluate this expression, we can use the linearity property of integrals, which states that the integral of a sum is equal to the sum of the integrals. Therefore, we can separate the integral ∫f(x) dx and ∫g(x) dx and compute them individually.
∫f(x) dx represents the integral of the function f(x) with respect to x. To evaluate this integral, we need to find the antiderivative of f(x), which is a function whose derivative is equal to f(x). We can use techniques such as integration by substitution, integration by parts, or partial fractions to find the antiderivative of f(x). Once we find the antiderivative, we add the constant of integration, which can be any arbitrary constant. The result of this integration will be denoted by F(x) + C.
∫g(x) dx represents the integral of the function g(x) with respect to x. We apply the same process we did with ∫f(x) dx- that is to find the antiderivative of g(x), denote it by G(x) + C, where C is the constant of integration.
Therefore, the final result will be the sum of the indefinite integrals of f(x) and g(x):
∫f(x) dx + ∫g(x) dx = F(x) + C + G(x) + C = F(x) + G(x) + 2C.
Alternatively, if we have definite integrals, we can evaluate them separately and then add their values:
∫f(x) dx + ∫g(x) dx = [F(b) – F(a)] + [G(b) – G(a)] = F(b) + G(b) – F(a) – G(a),
where a and b are the limits of integration.
More Answers:
Understanding ln |x| + C: Properties and Graphical Representation of Natural Logarithm of Absolute Value Function with a Constant CMaster the Power Rule of Integration: Indefinite Integrals of x^(n+1)/(n+1)
Mastering the Fundamental Theorem of Calculus: Evaluating the Expression ∫f(x) dx – ∫g(x) dx with Limits of Integration