Solve for: integral of (sin(x))/(cos(x)^2) x

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Expression: $\int{ \frac{ \sin\left({x}\right) }{ {\cos\left({x}\right)}^{2} } } \mathrm{d} x$

Use the substitution $t=\cos\left({x}\right)$ to transform the integral

$\int{ -\frac{ 1 }{ {t}^{2} } } \mathrm{d} t$

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

$-\int{ \frac{ 1 }{ {t}^{2} } } \mathrm{d} t$

Use $\begin{array} { l }\int{ \frac{ 1 }{ {x}^{n} } } \mathrm{d} x=-\frac{ 1 }{ \left( n-1 \right) \times {x}^{n-1} },& n≠1\end{array}$ to evaluate the integral

$-\left( -\frac{ 1 }{ t } \right)$

Substitute back $t=\cos\left({x}\right)$

$-\left( -\frac{ 1 }{ \cos\left({x}\right) } \right)$

Simplify the expression

$\frac{ 1 }{ \cos\left({x}\right) }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\frac{ 1 }{ \cos\left({x}\right) }+C,& C \in ℝ\end{array}$

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