# Trigonometric Integrals

### 1. Basic Trigonometric Identities:

• Sine and Cosine Identities:
• ${\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1$
• $\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$
• $\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}$, $\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right)}$

### 2. Trigonometric Integrals:

• Common Integrals:
• $\int \mathrm{sin}\left(x\right)\text{\hspace{0.17em}}dx=-\mathrm{cos}\left(x\right)+C$
• $\int \mathrm{cos}\left(x\right)\text{\hspace{0.17em}}dx=\mathrm{sin}\left(x\right)+C$
• $\int \mathrm{tan}\left(x\right)\text{\hspace{0.17em}}dx=-\mathrm{ln}\mathrm{\mid }\mathrm{cos}\left(x\right)\mathrm{\mid }+C$
• $\int \mathrm{sec}\left(x\right)\text{\hspace{0.17em}}dx=\mathrm{ln}\mathrm{\mid }\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)\mathrm{\mid }+C$
• $\int \mathrm{csc}\left(x\right)\text{\hspace{0.17em}}dx=-\mathrm{ln}\mathrm{\mid }\mathrm{csc}\left(x\right)+\mathrm{cot}\left(x\right)\mathrm{\mid }+C$

### 3. Trigonometric Substitution:

• Principle: Substitutes trigonometric expressions to simplify integrals involving radicals.
• Substitutions:
• For $\sqrt{{a}^{2}-{x}^{2}}$, use $x=a\mathrm{sin}\left(\theta \right)$
• For $\sqrt{{x}^{2}-{a}^{2}}$, use $x=a\mathrm{sec}\left(\theta \right)$
• For $\sqrt{{x}^{2}+{a}^{2}}$, use $x=a\mathrm{tan}\left(\theta \right)$
•

### 4. Trigonometric Identities and Techniques:

• Double Angle Formulas:

• $\mathrm{sin}\left(2x\right)=2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$
• $\mathrm{cos}\left(2x\right)={\mathrm{cos}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(x\right)$
• $\mathrm{tan}\left(2x\right)=\frac{2\mathrm{tan}\left(x\right)}{1-{\mathrm{tan}}^{2}\left(x\right)}$
• Half-Angle Formulas:

• ${\mathrm{sin}}^{2}\left(x\right)=\frac{1-\mathrm{cos}\left(2x\right)}{2}$
• ${\mathrm{cos}}^{2}\left(x\right)=\frac{1+\mathrm{cos}\left(2x\right)}{2}$

### Types of Trigonometric Integrals:

1. Basic Trigonometric Integrals:

• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• These are fundamental integrals involving sine and cosine functions.
2. Powers of Trigonometric Functions:

• $Si{n}^{n}$(x) $Co{s}^{m}$(x) dx
• Integrating powers of sine and cosine functions involves various techniques like trigonometric identities or reduction formulas.
3. Products of Trigonometric Functions:

• Integrals of products of trigonometric functions like sin(x)cos(x) or sin(x)sin(2x) can be solved using substitution or integration by parts.
4. Trigonometric Substitution:

• Substituting trigonometric expressions to simplify integrals involving radicals or square roots.
• Common substitutions include letting $x=a\mathrm{sin}\left(\theta \right)$ or $x=a\mathrm{tan}\left(\theta \right)$ to simplify the integral.

### Tips for Solving Trigonometric Integrals:

1. Trigonometric Identities:

• Utilize trigonometric identities to simplify integrals involving trigonometric functions.
• Examples include double-angle, half-angle, and Pythagorean identities.
2. Symmetry Properties:

• Exploit the symmetry properties of trigonometric functions (even or odd) to simplify integrals.
• Utilize periodicity properties to reduce the range of integration.
3. Integration by Parts:

• Helpful when dealing with products of trigonometric functions or functions that don't lend themselves to straightforward substitutions.
• Use the formula ∫ u dv = uv - ∫ v du.
4. Trigonometric Reduction Formulas:

• Useful for reducing powers of trigonometric functions into simpler expressions.
• Example: The reduction formula ∫ $Si{n}^{n}$(x) dx = -1/n * $Si{n}^{\left(n-1\right)}$(x)cos(x) + (n-1)/n * ∫ $Si{n}^{\left(n-2\right)}$(x) dx.

### Example:

Let's solve the indefinite integral ∫ $Si{n}^{2}$(x) dx:

### Solution:

Using the trigonometric identity ${\mathrm{sin}}^{2}\left(x\right)=\frac{1-\mathrm{cos}\left(2x\right)}{2}$, the integral becomes:

$\int {\mathrm{sin}}^{2}\left(x\right)dx=\frac{1}{2}\int \left(1-\mathrm{cos}\left(2x\right)\right)dx$

$\frac{1}{2}\left(x-\frac{1}{2}\mathrm{sin}\left(2x\right)\right)+C$

### Example:

Let's solve the indefinite integral ∫ sin(x) cos(x) dx:

### Solution:

To solve this integral, we'll use integration by parts. The formula for integration by parts is ∫ u dv = uv - ∫ v du.

Here, let's choose:

• $u=\mathrm{sin}\left(x\right)$ (so $du=\mathrm{cos}\left(x\right)dx$)
• $dv=\mathrm{cos}\left(x\right)dx$ (so $v=\mathrm{sin}\left(x\right)$)

Applying the integration by parts formula: $\int \mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx=\mathrm{sin}\left(x\right)\cdot \mathrm{sin}\left(x\right)-\int \mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right)dx$

Let's rearrange this equation to solve for the integral on the left side: $\int \mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx={\mathrm{sin}}^{2}\left(x\right)-\int \mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx$

Now, move the integral term to the right side: $2\int \mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx={\mathrm{sin}}^{2}\left(x\right)$ $\int \mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx=\frac{{\mathrm{sin}}^{2}\left(x\right)}{2}+C$