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ddd [48]
2 years ago
5

Evaluate the integral. (remember to use absolute values where appropriate. Use c for the constant of integration.) 5 cot5(θ) sin

4(θ) dθ
Mathematics
1 answer:
ozzi2 years ago
7 0

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

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THANK YOU , PLEASE INCLUDE EXPLANATION
frosja888 [35]

Remark

This will be a little late, but I'll answer it anyway. The more you see of the sine law, the clearer it will be.


Step One

Set up the sine Law.

\dfrac{Sin(x)}{12}=\dfrac{sin(75)}{22}


Step Two

Substitute values

Sin(x) = ??

sin(75) = 0.9659


\dfrac{Sin(x)}{12}=\dfrac{0.9659}{22}


Step Three

Cross multiply and Solve

22* Sin(x) = 12 * 0.9659

22* Sin(x) = 11.591 divide by 22

sin(x) = 11.591 / 22

sin(x) = 0.5269


Step Four

You have to know how to use the inverse of an angle to get the answer. I do it as my calculator would do it. If it doesn't work with yours, let me know.


x = sin^-1(0.5269)

2nd F

Sin^-1

(

0.5269

)

=

You should get 31.79 degrees.

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Answer:


Step-by-step explanation:

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