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Evaluate the value of the given integral \int cot \left ( 2x+1 \right )dx

Option: 1

\frac{1}{2}ln\left | cos\left ( 2x+1 \right ) \right |+C


Option: 2

\frac{1}{2}ln\left | cos\left ( 2x-1 \right ) \right |+C


Option: 3

\frac{1}{2}ln\left | sin\left ( 2x+1 \right ) \right |+C


Option: 4

\frac{1}{2}ln\left | sin\left ( 2x-1 \right ) \right |+C


Answers (1)

best_answer

Given integral,

I=\int cot \left ( 2x+1 \right )dx

I=\int \frac{cos\left ( 2x+1 \right )}{sin\left ( 2x+1 \right )}dx

Substitute,

t=sin\left (2x+1 \right )

dt=2cos\left (2x+1 \right )

Thus,

cos\left (2x+1 \right )=\frac{dt}{2}

Therefore,

I=\int\frac{dt}{2sin\ t}

I=\frac{1}{2}ln\left | t \right |+C

\int cot \left ( 2x+1 \right )dx=\frac{1}{2}ln\left | sin\left ( 2x+1 \right ) \right |+C

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Rishabh

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