$I=5\int \frac{cos^{4}\theta }{sin\theta }\times cos\theta d\theta \\\\I=5\int \left ( 1-sin^{2}\theta \right )^{2}\times \frac{cos\theta }{sin\theta }d\theta \\put\ \sin\theta =t\\\\dt=cos\theta d\theta \\\\I=5\int\frac{t^{4}+1-2t^{2}}{t}dt\ \ \ \ \ \ \ \ \ \ \because (a-b)^2=a^2+b^2-2ab\\\\I=5\left ( \int t^{3}dt + \int \frac{1}{t} -2\int t \right )dt$

by using the integration formula

we get,

$\\I=5\left ( \frac{t^{4}}{4} +logt -t^{2}\right )\\\\I=\frac{5}{4}t^{4}+5\log t-5t^{2}+c$

now put the value of t=\sin\theta in the above equation

we get,

$\int 5\cot^5\theta \sin^4\theta d\theta=\frac{5}{4}sin^{4}\theta+5\log \sin\theta - 5sin^{2} \theta+c$

hence proved

7 0

3 years ago

What two positive real numbers whose product is 50 have the smallest possible sum?

Llana [10]

X * y = 50
y = 50 / x
S = x + y = x + 50 / x
S` = 1 - 50 / x²
1 - 50 / x² = 0
50 / x² = 1
x² = 50
x = √ 50 = 5 √2
y = 50 / 5√2 = 5 √2
Answer:
Those numbers are: x = 5√2 and y = 5√2.

7 0

3 years ago

• 6 mystery books

Charra [1.4K]

Answer:

Step-by-step explanation:

How many books are there?

6+5+4+3 = 18 books total

How many are adventure books?

3 out of 18 books are adventure books

Robero has a 3/18 probability of choosing an adventure book.

Your teacher might want that probability in the lowest possible form. That is because it is the easiest to use and to understand.

You can do this by dividing both numbers by the "numerator" which is the first number in the fraction, (the adventure book count).

3/3 = 1

18/3 = 6

So the fraction can be reduced to 1/6.

Robero has a 1/6 chance of choosing an adventure book.

7 0

3 years ago

What is the length of side s of the square shown below? 8 90"

tankabanditka [31]

C bc pythagorean theorem

4 0

3 years ago

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