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Evaluate \int \csc x \cot ^{8}x dx

  • Option 1)

    \frac{\cot ^{9}x}{9}+C

  • Option 2)

    \frac{\cot ^{10}x}{10}+C

  • Option 3)

    \frac{\cot ^{8}x}{8}+C

  • Option 4)

    \frac{\cot ^{7}x}{7}+C

 

Answers (1)

As we have learned

Integration of trigonometric function of power m -

\int sin^{m}xdx , \int cos^{m}xdx, \int tan^{m}xdx :

 

 

 

- wherein

for m=4.

\therefore \int tan^{4}xdx =\int tan^{2}x\cdot tan^{2}xdx=\int\left ( sec^{2} x-1\right )tan^{2}xdxUse sin^{4}x=\left ( sin^{2}x \right )^{2}=\left (\frac{1-cos2x}{2} \right )^{2},cos^{4}x=\left ( cos^{2}x \right )^{2}=\left ( \frac{1+cos2x}{2} \right )^{2}

 

 \int csc^{2}x \cot ^{8}x dx

put \cot x = t ; \csc^{2}x dx = dt

\int \cot^{8}dt =\frac{t^{9}}{9}+ C= \frac{\cot^{9}x}{9}+C

 

 

 

 

 


Option 1)

\frac{\cot ^{9}x}{9}+C

This is incorrect

Option 2)

\frac{\cot ^{10}x}{10}+C

This is incorrect

Option 3)

\frac{\cot ^{8}x}{8}+C

This is correct

Option 4)

\frac{\cot ^{7}x}{7}+C

This is incorrect

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subam

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