18.2. Definite Integral of Trigonometric Functions - Graphing Calculator by Mathlab: User Manual

18.2. Definite Integral of Trigonometric Functions

This section will look at the integrals of trigonometric functions. Recall the properties of trigonometric integrals:

$1)\int \textup{sin}x\, dx=-\textup{cos}x+c$

$2)\int \textup{cos}x\, dx=\textup{sin}x+c$

$3)\int \textup{sec}^2x\, dx=\textup{tan}x+c$

$4)\int \textup{csc}^2x\, dx=-\textup{cot}x+c$

$5)\int \textup{sec}x\textup{tan}x\, dx=\textup{sec}x+c$

$6)\int \frac{1}{x^2+1}\: dx=\textup{tan}^{-1}x+c$

$7)\int \textup{csc}x\, \textup{cot}x\, dx=-\textup{csc}x+c$

$8)\int \textup{tan}x\, dx=\textup{ln}\, \left |\textup{sec}x \right |+c$

$9)\int \textup{cot}x\, dx=\textup{ln}\, \left |\textup{sin}x \right |+c$

$10)\int \textup{sec}x\, dx=\textup{ln}\, \left |\textup{sec}x +\textup{tan}x \right |+c$

$11)\int \textup{csc}x\, dx=\textup{ln}\, \left |\textup{csc}x -\textup{cot}x \right |+c$

Examples
Find each definite integral below.

Note: Enter degree symbol after typing the lower and upper limits.

$1)\int_{30^{\circ}}^{60^{\circ}}\left ( \textup{cos}x\, \textup{sin}^5x \right )\, dx$

Calculator syntax:

$\int [30^{\circ},60^{\circ}]\left ( \textup{cos}x\: \textup{sin}^5x \right )\textup{dx}$

$2)\int_{15^{\circ}}^{60^{\circ}}\left ( \textup{csc}\: x+3\textup{tan}x \right )\: dx$

Calculator syntax:

$\int [15^{\circ},60^{\circ}]\left ( \textup{csc}x\:+ 3\: \textup{tan}x \right )\textup{dx}$

$3)\int_{15^{\circ}}^{60^{\circ}}\left ( \textup{csc}\: x\: \textup{cot}x \right )\: dx$

Calculator syntax:

$\int [15^{\circ},60^{\circ}]\left ( \textup{csc}x\:\textup{cot}x \right )\textup{dx}$

$4)\int_{45^{\circ}}^{120^{\circ}}\left ( \textup{csc}^2x \right )\textup{dx}$

Calculator syntax:

$\int [45^{\circ},120^{\circ}]\left ( \textup{csc}^2x\:)\textup{dx}$

$5)\int_{45^{\circ}}^{90^{\circ}}\left ( \textup{cos}\; x -1\right )\textup{dx}$

Calculator syntax:

$\int[45^{\circ},90^{\circ}]( \textup{cos}\; x-1 \right ))\: dx$

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