Evaluate the following standard integral: [tex]$\int 4 \cos 3 x d x$[/tex]
Answer (2)
Perform a u-substitution by setting u = 3 x , which implies d x = 3 1 d u .
Rewrite the integral as 3 4 ∫ cos ( u ) d u .
Integrate to get 3 4 sin ( u ) + C .
Substitute back to obtain the final answer: 3 4 sin ( 3 x ) + C .
Explanation
Problem Analysis We are asked to evaluate the integral ∫ 4 cos 3 x d x . This is a standard integral that can be solved using a simple substitution.
U-Substitution Let's use the substitution method. Let u = 3 x . Then, the derivative of u with respect to x is d x d u = 3 . Therefore, d u = 3 d x , which means d x = 3 1 d u .
Substituting u and dx Now, substitute u and d x into the integral: ∫ 4 cos 3 x d x = ∫ 4 cos u ⋅ 3 1 d u = 3 4 ∫ cos u d u
Integrating The integral of cos u with respect to u is sin u . So, we have 3 4 ∫ cos u d u = 3 4 sin u + C where C is the constant of integration.
Substituting Back Finally, substitute back u = 3 x to get the result in terms of x : 3 4 sin u + C = 3 4 sin 3 x + C
Examples
Integrals of trigonometric functions are essential in physics and engineering. For example, when analyzing alternating current (AC) circuits, the current and voltage often vary sinusoidally with time. Calculating the average power dissipated in a resistor involves integrating the square of the current over a period. The integral of cos ( a t ) or sin ( a t ) appears frequently in such calculations, helping engineers determine the energy consumption and optimize circuit designs.
The integral ∫ 4 cos 3 x d x can be evaluated using substitution, resulting in 3 4 sin ( 3 x ) + C . We defined u = 3 x and computed the integral step-by-step. The final answer incorporates the constant of integration C .
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