integration of sinx + sin2x with limit 180/2 and 0
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We need to evaluate the definite integral of sine x plus sine 2x from 0 to 90 degrees. First, let's visualize the functions. The blue curve shows sine x, the red curve shows sine 2x, and the green curve shows their sum. The yellow area represents the integral we need to calculate.
First, we need to convert our integration limits from degrees to radians. The upper limit is 180 divided by 2, which equals 90 degrees. Converting to radians: 90 degrees equals 90 times pi over 180, which simplifies to pi over 2 radians. The lower limit is 0 degrees, which equals 0 radians. So our integral becomes the integral from 0 to pi over 2 of sine x plus sine 2x dx.
Now we find the indefinite integral of sine x plus sine 2x. We can split this into two separate integrals. The integral of sine x is negative cosine x plus a constant. For the integral of sine 2x, we use substitution. Let u equal 2x, then du equals 2 dx, so dx equals du over 2. This gives us negative one-half cosine 2x plus a constant. Combining both results, the indefinite integral is negative cosine x minus one-half cosine 2x plus C.
Now we apply the Fundamental Theorem of Calculus. We evaluate our antiderivative at the upper and lower limits, then subtract. At the upper limit x equals pi over 2: cosine of pi over 2 equals 0, and cosine of pi equals negative 1. At the lower limit x equals 0: cosine of 0 equals 1. The unit circle helps us visualize these cosine values at key angles.
Now we substitute the cosine values and perform the final calculation. We have negative 0 minus one-half times negative 1, minus the quantity negative 1 minus one-half times 1. This simplifies to one-half minus negative three-halves, which equals one-half plus three-halves, giving us four-halves, which equals 2. Therefore, the definite integral of sine x plus sine 2x from 0 to pi over 2 equals 2.