Q. 0 - Calculus

Question

read the section and make your own summary of material.

Step-by-Step Solution

Answer

 The chapter defines the indefinite integrals and states the formulas for Integrals of Exponential Functions, Certain Trigonometric Expressions, Constant Multiple and Sum Rules for Indefinite Integrals, and Reversing the Product, Quotient, and Chain Rules.

Step 1. Given Information.

The topic of the given section is Indefinite Integrals.

Step 2. definations.

Functions with the Same Derivative Differ by a Constant.

If functions f and g are differentiable then $f\text{'}\left(x\right)=g\text{'}\left(x\right)$when $f\left(x\right)=g\left(x\right)+C$for some constant C.

Indefinite Integral.

An indefinite Integral is a family of functions that differ by a constant.

$\int f\left(x\right) dx=F\left(x\right)+C.$

where F is the antiderivative of f so, $F\text{'}=f.$

Step 2. Formulas.

Integrals of Power Functions.

$$\begin{gathered} \left(i\right) \int x^k dx=\frac{1}{k+1}{x}^{k+1}+C, \mathrm{Where} k\ne -1 \\ \left(ii\right) \int \frac{1}{x}dx=\ln \left|x\right|+C \end{gathered}$$

Integrals of Exponential Functions.

$$\begin{gathered} \left(i\right) \int e^{kx}dx=\frac{1}{k}{e}^{kx}+C, \mathrm{Where} k\ne 0. \\ \left(ii\right) \int b^{x}dx=\frac{1}{\ln b}{b}^x+C, \mathrm{where} b>1. \end{gathered}$$

Integrals of Certain Trigonometric Expressions.

$$\begin{gathered} \left(i\right) \int \sin \left(x\right) dx=-\cos x+ C \\ \left(ii\right) \int \cos x dx=\sin x+ C \\ \left(iii\right) \int se{c}^{2}x dx=\tan x+ C \\ \left(iv\right) \int cs{c}^{2 }x dx=-\mathrm{co}t x+ C \\ \left(v\right) \int sec x \tan x dx=sec x+ C \\ \left(vi\right) \int csc x cot x dx=-csc x+ C \end{gathered}$$

Constant Multiple and Sum Rules for Indefinite Integrals.

$$\begin{gathered} \left(i\right) \int kf\left(x\right) dx=k\int f\left(x\right) dx \\ \left(ii\right) \int \left[f\left(x\right)+g\left(x\right)\right] dx=\int f\left(x\right) dx+\int g\left(x\right) dx \end{gathered}$$

Reversing the Product, Quotient, and Chain Rules.

$$\begin{gathered} \int \left[f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)\right] dx=f\left(x\right) g\left(x\right) +C \\ \int \frac{\left[f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)\right] }{{\left[g\left(x\right)\right]}^2}dx=\frac{f\left(x\right) }{g\left(x\right) }+C \\ \int \left[f\text{'}\left(g\right(x\left)\right)g\text{'}\left(x\right)\right] dx=f\left(g\right(x) )+C \end{gathered}$$