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Evalute \int \csc ^{4}xdx

  • Option 1)

    \cot x + \cot ^{3} x /3 +C

  • Option 2)

    \cot ^{2}x +\cot ^{4}x /4 + C

  • Option 3)

    1+ \cot ^{2}x /2 +C

  • Option 4)

    \frac{\csc ^{5}x}{5} +C

 

Answers (1)

best_answer

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 ^{4}xdx

=\int csc ^{2}x \cdot \csc ^{2}xdx= \int (1+ \cot^{2}x) \csc ^{2} xdx

\cot x + \frac{\cot ^{3}x}{3} + C 

 

 

 

 

 

 


Option 1)

\cot x + \cot ^{3} x /3 +C

This is correct

Option 2)

\cot ^{2}x +\cot ^{4}x /4 + C

This is incorrect

Option 3)

1+ \cot ^{2}x /2 +C

This is incorrect

Option 4)

\frac{\csc ^{5}x}{5} +C

This is incorrect

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