Use the symmetry of the graphs of the sine and cosine functions as an aid in evaluating each definite integral.

(a) Integral of sinx*dx from -pi/4 to pi/4

(b) Integral of cosx*dx from -pi/4 to pi/4

(c) Integral of cosx*dx from -pi/2 to pi/2

(d) Integral of sinx*cosx*dx from -pi/2 to pi/2

To use the symmetry of the graphs of the sine and cosine functions in evaluating definite integrals, we need to understand their characteristics.

Both the sine and cosine functions are periodic with a period of 2π. This means that their graphs repeat every 2π units.

The sine function is an odd function, which means that it is symmetric with respect to the origin (0,0). In other words, if we reflect the graph of the sine function across the y-axis or x-axis, it will be unchanged.

The cosine function, on the other hand, is an even function. This means that it is symmetric with respect to the y-axis. If we reflect the graph of the cosine function across the y-axis, it will be unchanged.

Now let's evaluate each definite integral using the symmetry of these functions:

(a) ∫ sin(x) dx from -π/4 to π/4:

Since the sine function is odd, we can use the symmetry across the y-axis to simplify the integral. The integral from -π/4 to π/4 will give us an area of zero, because the positive and negative areas will cancel each other out.

So the value of this integral is 0.

(b) ∫ cos(x) dx from -π/4 to π/4:

The cosine function is an even function, so we can use the symmetry across the y-axis to simplify the integral. The positive and negative areas will be equal, so the value of this integral will be twice the integral from 0 to π/4.

Since the integral of cos(x) from 0 to π/4 is sin(π/4) - sin(0) = (sqrt(2)/2) - 0 = sqrt(2)/2, the value of this integral is 2 * (sqrt(2)/2) = sqrt(2).

(c) ∫ cos(x) dx from -π/2 to π/2:

By using the symmetry of the cosine function, we know that the positive and negative areas will be equal, so the value of this integral is twice the integral from 0 to π/2.

Since the integral of cos(x) from 0 to π/2 is sin(π/2) - sin(0) = 1 - 0 = 1, the value of this integral is 2 * 1 = 2.

(d) ∫ sin(x) * cos(x) dx from -π/2 to π/2:

To evaluate this integral, we can use the odd-even property. Since sin(x) is an odd function and cos(x) is an even function, the product sin(x) * cos(x) is an odd function.

As an odd function, the graph of sin(x) * cos(x) is symmetric with respect to the origin. This means that the positive and negative areas will be equal, so the value of this integral is zero.

Therefore, the value of this integral is 0.