Find the antiderivative of csc(y)dx+sec(x)dy

 Exercise

$\csc\left(y\right)dx+\sec\left(x\right)dy$

 

Find the integral

$\int\csc\left(y\right)dx+\int\sec\left(x\right)dy$

 

The integral $\int\csc\left(y\right)dx$ results in: $-\ln\left(\csc\left(y\right)+\cot\left(y\right)\right)$

$-\ln\left(\csc\left(y\right)+\cot\left(y\right)\right)$

 

The integral $\int\sec\left(x\right)dy$ results in: $\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$

$\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$

 

Gather the results of all integrals

$-\ln\left|\csc\left(y\right)+\cot\left(y\right)\right|+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|$

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$-\ln\left|\csc\left(y\right)+\cot\left(y\right)\right|+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$

$-\ln\left|\csc\left(y\right)+\cot\left(y\right)\right|+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$

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