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

$=\dfrac{1}{2}\left(\dfrac{1}{3}\sin^3(2x)-\dfrac{1}{5}\sin^5(2x)\right)+C$

4 0

3 years ago

How to determine if set of vectors is linearly independent?

zimovet [89]

The vectors $\{v_1,v_2,...,v_p\}\in \mathbb{R}^n$ are linearly independent if the ONLY solution to the following equation:

$c_1v_1+c_2v_2+...c_pv_p=0$

is $c_1=c_2=...=c_p=0$

7 0

3 years ago

The sum of three numbers is 108. The second number is four times greater than the first number, and the third number is 18 more

tankabanditka [31]

Answer:

450

Step-by-step explanation:

108 x 4 +18

6 0

2 years ago