Evaluate: integral of 3csc(t)^2-5sec(t)tan(t) t

Expression: $\int{ 3{\csc\left({t}\right)}^{2}-5\sec\left({t}\right)\tan\left({t}\right) } \mathrm{d} t$

Use $\csc\left({t}\right)=\frac{ 1 }{ \sin\left({t}\right) }$ to transform the expression

$\int{ 3 \times {\left( \frac{ 1 }{ \sin\left({t}\right) } \right)}^{2}-5\sec\left({t}\right)\tan\left({t}\right) } \mathrm{d} t$

Use $\sec\left({t}\right)=\frac{ 1 }{ \cos\left({t}\right) }$ to transform the expression

$\int{ 3 \times {\left( \frac{ 1 }{ \sin\left({t}\right) } \right)}^{2}-5 \times \frac{ 1 }{ \cos\left({t}\right) } \times \tan\left({t}\right) } \mathrm{d} t$

Use $\tan\left({t}\right)=\frac{ \sin\left({t}\right) }{ \cos\left({t}\right) }$ to transform the expression

$\int{ 3 \times {\left( \frac{ 1 }{ \sin\left({t}\right) } \right)}^{2}-5 \times \frac{ 1 }{ \cos\left({t}\right) } \times \frac{ \sin\left({t}\right) }{ \cos\left({t}\right) } } \mathrm{d} t$

To raise a fraction to a power, raise the numerator and denominator to that power

$\int{ 3 \times \frac{ 1 }{ {\sin\left({t}\right)}^{2} }-5 \times \frac{ 1 }{ \cos\left({t}\right) } \times \frac{ \sin\left({t}\right) }{ \cos\left({t}\right) } } \mathrm{d} t$

Calculate the product

$\int{ 3 \times \frac{ 1 }{ {\sin\left({t}\right)}^{2} }-\frac{ 5\sin\left({t}\right) }{ {\cos\left({t}\right)}^{2} } } \mathrm{d} t$

Calculate the product

$\int{ \frac{ 3 }{ {\sin\left({t}\right)}^{2} }-\frac{ 5\sin\left({t}\right) }{ {\cos\left({t}\right)}^{2} } } \mathrm{d} t$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\int{ \frac{ 3 }{ {\sin\left({t}\right)}^{2} } } \mathrm{d} t-\int{ \frac{ 5\sin\left({t}\right) }{ {\cos\left({t}\right)}^{2} } } \mathrm{d} t$

Evaluate the indefinite integral

$-3\cot\left({t}\right)-\int{ \frac{ 5\sin\left({t}\right) }{ {\cos\left({t}\right)}^{2} } } \mathrm{d} t$

Evaluate the indefinite integral

$-3\cot\left({t}\right)-\frac{ 5 }{ \cos\left({t}\right) }$

Simplify the expression

$-\frac{ 3\cos\left({t}\right) }{ \sin\left({t}\right) }-\frac{ 5 }{ \cos\left({t}\right) }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }-\frac{ 3\cos\left({t}\right) }{ \sin\left({t}\right) }-\frac{ 5 }{ \cos\left({t}\right) }+C,& C \in ℝ\end{array}$