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  4. The value of the definite integral ∫ limitsθ1θ2 ( d θ/1+ tan θ) where θ2=(1004 π/2010) and θ1=(π/2010) is equal to

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Q. The value of the definite integral $\int\limits_{\theta_1}^{\theta_2} \frac{ d \theta}{1+\tan \theta}$ where $\theta_2=\frac{1004 \pi}{2010}$ and $\theta_1=\frac{\pi}{2010}$ is equal to

Integrals

Solution:

$I =\int\limits_{\theta_1}^{\theta_2} \frac{ d \theta}{1+\tan \theta} ; \theta_1+\theta_2=\frac{\pi}{2}$
$I=\int\limits_{\theta_1}^{\theta_2} \frac{d \theta}{1+\cot \theta}=\int\limits_{\theta_1}^{\theta_2} \frac{\tan \theta d \theta}{1+\tan \theta} $ (using King)
$2 I=\int\limits_{\theta_1}^{\theta_2} d \theta=\theta_2-\theta_1=\frac{1003 \pi}{2010} \Rightarrow I=\frac{1003 \pi}{4020}$