Calculus: Integrals III

Trigonometric Functions

The integrals of the three basic trig functions are as follows:

$$\int \sin(x) \, dx = -\cos(x) + C$$

$$\int \cos(x) \, dx = \sin(x) + C$$

$$\int \tan(x) \, dx = -\ln|\cos(x)| + C = \ln|\sec(x)| + C$$

The integrals of sine and cosine follow easily from their derivatives. You'll be able to solve for the integral of tangent directly in the section on $u$-substitution. Since we know the derivatives of $\tan(x)$ and $\cot(x)$, we also can derive the following two integrals:

$$\int \sec^2(x) \, dx = \tan(x) + C$$

$$\int \csc^2(x) \, dx = -\cot(x) + C$$

Problems

  1. Evaluate: $\displaystyle\int \sqrt{1-\cos^2(x)} \, dx$

    $\displaystyle\int \sqrt{1-\cos^2(x)} \, dx = \displaystyle\int \sqrt{\sin^2(x)} \, dx \\ \displaystyle\int \sqrt{1-\cos^2(x)} \, dx = \displaystyle\int |\sin(x)| \, dx \\ \displaystyle\int \sqrt{1-\cos^2(x)} \, dx = \left\{ \begin{array}[ll] \phantom{}-\cos(x) & 2k\pi<x<2(k+1)\pi \\ \cos(x) & 2(k+1)\pi<x<2(k+2)\pi \\ \end{array} \right. \text{ where } k \in \mathbb{Z} \\
    $

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  2. Evaluate: $\displaystyle\int \cot(\theta)\sin(\theta) \, d\theta$

    $\displaystyle\int \cot(\theta)\sin(\theta) \, d\theta = \displaystyle\int \dfrac{\cos(\theta)}{\sin(\theta)}\sin(\theta) \, d\theta \\ \displaystyle\int \cot(\theta)\sin(\theta) \, d\theta = \displaystyle\int \cos(\theta) \, d\theta \\ \displaystyle\int \cot(\theta)\sin(\theta) \, d\theta = \sin(\theta) + c \\ $

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  3. Evaluate $\displaystyle\int \dfrac{\sin(2x)}{\sin(x)} \, dx$

    $\displaystyle\int \dfrac{\sin(2x)}{\sin(x)} \, dx = \displaystyle\int \dfrac{2\sin(x)\cos(x)}{\sin(x)} \, dx \\ \displaystyle\int \dfrac{\sin(2x)}{\sin(x)} \, dx = \displaystyle\int 2\cos(x) \, dx \\ \displaystyle\int \dfrac{\sin(2x)}{\sin(x)} \, dx = 2\sin(x) + c \\ $

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  4. Evaluate: $\displaystyle\int \dfrac{\tan(x)\cos^3(x)}{\sin(2x)} \, dx$

    $\displaystyle\int \dfrac{\tan(x)\cos^3(x)}{\sin(2x)} \, dx = \displaystyle\int \dfrac{\tan(x)\cos^3(x)}{2\sin(x)\cos(x)} \, dx \\ \displaystyle\int \dfrac{\tan(x)\cos^3(x)}{\sin(2x)} \, dx = \displaystyle\int \dfrac{\sin(x)\cos^3(x)}{2\sin(x)\cos^2(x)} \, dx \\ \displaystyle\int \dfrac{\tan(x)\cos^3(x)}{\sin(2x)} \, dx = \displaystyle\int \dfrac{1}{2}\cos(x) \, dx \\ \displaystyle\int \dfrac{\tan(x)\cos^3(x)}{\sin(2x)} \, dx = \dfrac{1}{2}\sin(x) + c \\ $

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  5. Evaluate: $\displaystyle\int \dfrac{\cos(2x) + \sin^2(x)}{\cos(x)} \, dx$

    $\displaystyle\int \dfrac{\cos(2x) + \sin^2(x)}{\cos(x)} \, dx = \displaystyle\int \dfrac{\cos^2(x) - \sin^2(x) + \sin^2(x)}{\cos(x)} \, dx \\ \displaystyle\int \dfrac{\cos(2x) + \sin^2(x)}{\cos(x)} \, dx = \displaystyle\int \dfrac{\cos^2(x)}{\cos(x)} \, dx \\ \displaystyle\int \dfrac{\cos(2x) + \sin^2(x)}{\cos(x)} \, dx = \displaystyle\int \cos(x) \, dx \\ \displaystyle\int \dfrac{\cos(2x) + \sin^2(x)}{\cos(x)} \, dx = \sin(x) + c \\ $

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