Find an anti derivative (or integral) of the following functions by the method of inspection: sin 2x
To find the integral of sin 2x, we refer to the integration methods shown in the provided text. The context demonstrates that the integral of sin 5x is -1/5 cos 5x, and the integral of sin 3x follows a similar pattern resulting in a cosine term divided by the coefficient.
Applying this observed rule for ∫ sin ax dx where the result is -1/a cos ax, we substitute the specific coefficient for our function. Here, the coefficient of x is 2. Thus, the integration yields: ∫ sin 2x dx = -1/2 cos 2x + C.
Explanation
The textbook context provides the solution for ∫ sin 5x dx as -1/5 cos 5x within the example for ∫ sin 2x cos 3x dx. Similarly, it shows the integration of sin 3x results in -1/3 cos 3x. By following this consistent pattern established in the text for integrating sine functions with a coefficient of x, the integral of sin 2x is derived by dividing the cosine function by the coefficient 2 and adding the constant of integration.
Solution Steps
Step 1: Identify the integral to be found: ∫ sin 2x dx.
Step 2: Refer to the context pattern: ∫ sin 5x dx = -1/5 cos 5x.
Step 3: Apply the pattern to the current function, dividing by the coefficient of x (which is 2).
Step 4: The final result is -1/2 cos 2x + C.