1) $\frac{\text{d}}{\text{d}x}sin(x) = cos(x) \Rightarrow \int_{}^{}cos(x)dx = sin(x) + c $
2) $\frac{\text{d}}{\text{d}x}cos(x) = -sin(x) \Rightarrow \int_{}^{}sin(x)dx = -cos(x) + c $
3) $\frac{\text{d}}{\text{d}x}tan(x) = sec^{2}(x) \Rightarrow \int_{}^{}sec^{2}(x)dx = tan(x) + c $
4) $\frac{\text{d}}{\text{d}x}cot(x) = -csc^{2}(x) \Rightarrow \int_{}^{}csc^{2}(x)dx = -cot(x) + c $
5) $\frac{\text{d}}{\text{d}x}sec(x) = sec(x)tan(x) \Rightarrow \int_{}^{}sec(x)tan(x)dx = sec(x) + c $
6) $\frac{\text{d}}{\text{d}x}csc(x) = -csc(x)cot(x) \Rightarrow \int_{}^{}csc(x)cot(x)dx = -csc(x) + c $
Example 1 : Integrate 2sin(x) with respect to x.
Solution : $\int_{}^{}2sin(x)dx$
= 2$ \int_{}^{}sin(x)dx$
= -2cos(x)
$\int_{}^{}2sin(x)dx $= -2cos(x) + c
Example 2 : Integrate $ \frac{1}{2}sec^{2}(x)$ with respect to x.
Solution : $\int_{}^{}\frac{1}{2}sec^{2}(x)dx $
=$ \frac{1}{2} \int_{}^{}sec^{2}(x)dx$
=$\frac{1}{2} tan(x)$
$\int_{}^{}\frac{1}{2}sec^{2}(x)dx = \frac{1}{2} tan(x) + c $
Example 3 : Integrate $ \frac{2}{1+cos(2x)}$ with respect to x.
Solution : $\int_{}^{}\frac{2}{1+cos(2x)}dx $
=$ \int_{}^{}\frac{2}{2cos^{2}(x)}dx$ ( Using the trigonometric identity)
=$ \int_{}^{}sec^{2}(x)dx$
= tan(x)
$\int_{}^{}\frac{2}{1+cos(2x)}dx = tan(x) + c $
Example 4: Integrate $ \frac{cos(2x) + 2sin^{2}(x)}{cos^{2}(x)}$ with respect to x.
Solution : $\int_{}^{}\frac{cos(2x) + 2sin^{2}(x)}{cos^{2}(x)}dx $
=$ \int_{}^{}\frac{1-2sin^{2}(x) + 2sin^{2}(x)}{cos^{2}(x)}dx$ ( Using the trigonometric identity)
=$ \int_{}^{}\frac{1}{cos^{2}(x)}dx$
=$ \int_{}^{}sec^{2}(x)dx$
= tan(x)
$\int_{}^{}\frac{cos(2x) + 2sin^{2}(x)}{cos^{2}(x)}dx = tan(x) + c $
Example 5 : Integrate $(x^{2}+4sin(x))$ with respect to x.
Solution : $\int_{}^{}(x^{2}+4sin(x))dx$
= $ \int_{}^{}x^{2}dx + \int_{}^{}4sin(x)dx$
= $ \int_{}^{}x^{2}dx + 4\int_{}^{}sin(x)dx$
= $ \frac{x^{(2+1)}}{2+1} - 4cos(x)$ (using the power rule of integration)
= $ \frac{x^{3}}{3} - 4cos(x)$
$\int_{}^{}(x^{2}+4sin(x)dx = \frac{x^{3}}{3} - 4cos(x) + c $