Integral of csc(x)\csc{\left(x \right)}

The calculator will find the integral/antiderivative of csc(x)\csc{\left(x \right)}, with steps shown.

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Find csc(x)dx\int \csc{\left(x \right)}\, dx.

Solution

Rewrite the cosecant as csc(x)=1sin(x)\csc\left(x\right)=\frac{1}{\sin\left(x\right)}:

csc(x)dx=1sin(x)dx{\color{red}{\int{\csc{\left(x \right)} d x}}} = {\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}

Rewrite the sine using the double angle formula sin(x)=2sin(x2)cos(x2)\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right):

1sin(x)dx=12sin(x2)cos(x2)dx{\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}

Multiply the numerator and denominator by sec2(x2)\sec^2\left(\frac{x}{2} \right):

12sin(x2)cos(x2)dx=sec2(x2)2tan(x2)dx{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}} = {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}}

Let u=tan(x2)u=\tan{\left(\frac{x}{2} \right)}.

Then du=(tan(x2))dx=sec2(x2)2dxdu=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx (steps can be seen »), and we have that sec2(x2)dx=2du\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du.

So,

sec2(x2)2tan(x2)dx=1udu{\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}

The integral of 1u\frac{1}{u} is 1udu=ln(u)\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}:

1udu=ln(u){\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}

Recall that u=tan(x2)u=\tan{\left(\frac{x}{2} \right)}:

ln(u)=ln(tan(x2))\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)}

Therefore,

csc(x)dx=ln(tan(x2))\int{\csc{\left(x \right)} d x} = \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}

Add the constant of integration:

csc(x)dx=ln(tan(x2))+C\int{\csc{\left(x \right)} d x} = \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}+C

Answer: csc(x)dx=ln(tan(x2))+C\int{\csc{\left(x \right)} d x}=\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}+C