Question

Integration of sin^2 2x cos^3 2x dx

Answer 1

To evaluate the given integral, we can use the power-reduction formulas for sine and cosine. The integral becomes a combination of cosines with different arguments, and we can integrate each term separately to find the final answer.

Step-by-step explanation:

To evaluate the integral ∫sin²(2x)cos³(2x)dx, we can use the power-reduction formulas for sine and cosine. The power-reduction formula for sine is sin²(x) = 1/2(1 - cos(2x)), and the power-reduction formula for cosine is cos³(x) = 1/4(3cos(x) + cos(3x)).

Replacing sin²(2x) and cos³(2x) with their respective power-reduction formulas, the integral becomes ∫(1/2)(1 - cos(4x)) * (1/4)(3cos(2x) + cos(6x))dx.

Expanding and simplifying the expression, we get ∫(3/8)cos(2x) - (1/8)cos(4x) - (1/32)cos(6x)dx. Now, we can integrate each term separately to get the final answer.

Answer 2

`\int\left[\sin^2(2x)\cos^3(2x)\right]dx=\int\left[\sin^2(2x)\cos^2(2x)\cos(2x)\right]dx\\\\\text{Use the trigonometric identity}\sin^2(x)+\cos^2(x)=1\to\cos^2(x)=1-\sin^2(x)\\\\=\int\left\{\sin^2(2x)[1-\sin^2(2x)]\cos(2x)\right\}dx\\\\\text{substitute}\ \sin2x=t\ \to\ 2\cos2x\ dx=dt\ \to\ \cos2x\ dx=(1)/(2)\ dt\\\\=(1)/(2)\int\left[t^2(1-t^2)\right]dt=(1)/(2)\int(t^2-t^4)dt=(1)/(2)\left((1)/(3)t^3-(1)/(5)t^5\right)+C`

`=(1)/(2)\left((1)/(3)\sin^3(2x)-(1)/(5)\sin^5(2x)\right)+C`