An indefinite integral is a mathematical operation that represents the antiderivative of a function. It is the reverse process of taking the derivative, and it involves finding a function whose derivative is the given function. It is usually denoted by the symbol ∫.
The process of solving an indefinite integral involves using the rules of integration, such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the antiderivative of a given function. In some cases, integration by substitution or integration by parts may also be necessary.
A definite integral has specific limits of integration, whereas an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral will give a general solution in terms of a constant of integration.
The process of solving an indefinite integral with trigonometric functions involves using the trigonometric identities and integration rules specific to trigonometric functions. In some cases, rewriting the trigonometric function in terms of a different trigonometric function or using integration by parts may also be necessary.
One way to solve this indefinite integral is to use the substitution method. Let u = 6x, then du = 6dx. Substituting this into the integral, we get:
∫ 5*(sin(6x)/sin(3x)) dx = ∫ 5*(sin(u)/sin(3x)) (du/6) = (5/6) ∫ (sin(u)/sin(3x)) du
Using the trigonometric identity sin(2x) = 2sin(x)cos(x), we can rewrite the integral as:
(5/6) ∫ (sin(u)/sin(3x)) du = (5/6) ∫ (2sin(u/2)cos(u/2))/(sin(3x)) du
Next, using the trigonometric identity sin(x)/sin(y) = 2cos((x-y)/2)sin((x+y)/2), we can further simplify the integral as:
(5/6) ∫ (2sin(u/2)cos(u/2))/(sin(3x)) du = (5/6) ∫ (2cos((u-3x)/2)sin((u+3x)/2)) du
Finally, using the substitution method again, let v = (u+3x)/2, then dv = (1/2)du. Substituting this into the integral, we get:
(5/6) ∫ (2cos((u-3x)/2)sin((u+3x)/2)) du = (5/6) ∫ 2cos(v) dv = (5/3) sin((u+3x)/2) + C
Replacing u with 6x, we get the final solution:
∫ 5*(sin(6x)/sin(3x)) dx = (5/3) sin((6x+3x)/2) + C = (5/3) sin(4.5x) + C