8.2.2 Integrating Other Functions (Trig, ln & e etc)

Integrating functions like trigonometric functions, the natural logarithm, and exponential functions is a crucial aspect of calculus. Here's a summary of how to integrate these functions, along with key formulas and examples.

1. Integrating Trigonometric Functions:

Basic Trigonometric Integrals:

1. Sine:

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

2. Cosine:

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

3. Secant Squared:

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

4. Cosecant Squared:

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

5. Secant-Tangent:

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

6. Cosecant-Cotangent:

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

Examples:

2. Integrating Exponential Functions:

The exponential function $e^x$ and its variants are straightforward to integrate.

7. Exponential Function $e^x :$

$\int e^x \, dx = e^x + C$

8. General Exponential Function $e^{kx} :$

$\int e^{kx} \, dx = \frac{1}{k}e^{kx} + C$

where $k$ is a constant.

Examples:

3. Integrating the Natural Logarithm Function:

The natural logarithm function $\ln(x)$ has a specific integration formula:

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

This formula can be derived using integration by parts.

Example:

4. Integration Involving Trigonometric Identities:

Sometimes, trigonometric identities simplify the integration process. For example:

5. Integrating Powers of Trigonometric Functions:

6. Integrating Functions Involving Inverse Trigonometric Functions:

1. Arcsine:

$\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin(x) + C$

2. Arctangent:

$\int \frac{1}{1 + x^2} \, dx = \arctan(x) + C$

3. Arcsecant:

$\int \frac{1}{x\sqrt{x^2 - 1}} \, dx = \ arcsec(x) + C$

Summary:

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Standard Integrals

There are standard integrals that you are expected to "spot" and quickly integrate with minimal working.

e.g.

• $(ax + b)^n$
• $\sin(ax + b), \cos(ax + b)$
• $e^{ax+b}$ The rule is: "integrate the full thing, divide by differential of bracket".

THIS ONLY WORKS WHEN THE BRACKET IS OF THE FORM $ax + b$.

(Reversing differentiation using formula booklet)

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Standard Integrals Summary:

1. $\int e^x \, dx = e^x + c$
2. $\int x^{-1} \, dx = \ln|x| + c$

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