Integration of Trigonometric Function

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In this section we will discuss the integration of trigonometric function.
Recall the definitions of the trigonometric functions and trigonometric identities.
1) $tan(x) = \frac{sin(x)}{Cos(x)}$

2) $cot(x) = \frac{cos(x)}{sin(x)}$

3) $sec(x) = \frac{sin(x)}{Cos(x)}$

4) $csc(x) = \frac{1}{sin(x)}$

5) $sin^{2}(x) + cos^{2}(x)=1$

6) $1 + tan^{2}(x)=sec^{2}(x)$

7) $1 + cot^{2}(x)=csc^{2}(x)$

8) sin(2x) = 2 sin(x) cos(x)

9) cos(2x) = $2cos^{2}(x) - 1 \Rightarrow cos^{2}(x)=\frac{1 + cos(2x)}{2}$

10) cos(2x) = $1-2sin^{2}(x) - 1 \Rightarrow sin^{2}(x)=\frac{1 - cos(2x)}{2}$

11) cos(2x) = $ cos^{2}(x) - sin^{2}(x)$

You have already learnt the derivatives of trigonometric functions. Integration is anti-differentiation.

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 $

Examples on Integration of Trigonometric Function

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 $

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