∫ sin x dx = -cos x + c; ∫ cos x dx = sin x + c; ∫ sec² x dx = tan x + c; ∫ cosec² x dx = -cot x + c; ∫ sec x tan x dx = sec x + c; ∫ 1/(√(x²-a²)) dx = (1/a) sec⁻¹(x/a) + c; ∫ -1/(... ∫ sin x dx = -cos x + c; ∫ cos x dx = sin x + c; ∫ sec² x dx = tan x + c; ∫ cosec² x dx = -cot x + c; ∫ sec x tan x dx = sec x + c; ∫ 1/(√(x²-a²)) dx = (1/a) sec⁻¹(x/a) + c; ∫ -1/(√(x²-a²)) dx = (1/a) cos⁻¹(x/a) + c; Some Important Results: f(ax+b) dx = (1/a) ϕ(ax+b)
Understand the Problem
The question consists of various integral formulas and important results related to calculus. It provides examples of integrals for different trigonometric functions and some general results on integration.
Answer
The key integrals are: 1. \( \int \sin x \, dx = -\cos x + c \) 2. \( \int \cos x \, dx = \sin x + c \) 3. \( \int \sec^2 x \, dx = \tan x + c \) 4. \( \int \csc^2 x \, dx = -\cot x + c \) 5. \( \int \sec x \tan x \, dx = \sec x + c \) The general result is \( \int f(ax + b) \, dx = \frac{1}{a} F(ax + b) + c \).
Answer for screen readers
The integrals listed are:
- ( \int \sin x , dx = -\cos x + c )
- ( \int \cos x , dx = \sin x + c )
- ( \int \sec^2 x , dx = \tan x + c )
- ( \int \csc^2 x , dx = -\cot x + c )
- ( \int \sec x \tan x , dx = \sec x + c )
The general integral result is: $$ \int f(ax + b) , dx = \frac{1}{a} F(ax + b) + c $$
Steps to Solve
- Integration of Sine Function
To find the integral of the sine function, we use the identity: $$ \int \sin x , dx = -\cos x + c $$
- Integration of Cosine Function
For finding the integral of the cosine function, we have: $$ \int \cos x , dx = \sin x + c $$
- Integration of Secant Squared Function
The integral for secant squared is given by: $$ \int \sec^2 x , dx = \tan x + c $$
- Integration of Cosecant Squared Function
Using the identity for cosecant squared, we write: $$ \int \csc^2 x , dx = -\cot x + c $$
- Integration of Secant and Tangent Function
The integral of secant multiplied by tangent is expressed as: $$ \int \sec x \tan x , dx = \sec x + c $$
- General Result for Linear Functions in Integrals
A key result for integrating linear transformations is: $$ \int f(ax + b) , dx = \frac{1}{a} F(ax + b) + c $$ where ( F ) is the antiderivative of ( f ).
The integrals listed are:
- ( \int \sin x , dx = -\cos x + c )
- ( \int \cos x , dx = \sin x + c )
- ( \int \sec^2 x , dx = \tan x + c )
- ( \int \csc^2 x , dx = -\cot x + c )
- ( \int \sec x \tan x , dx = \sec x + c )
The general integral result is: $$ \int f(ax + b) , dx = \frac{1}{a} F(ax + b) + c $$
More Information
These integrals are fundamental in calculus and are widely used in solving problems involving trigonometric functions. Understanding these integrals helps in tackling more complex integration problems.
Tips
- Confusing the integrals of sine and cosine functions; remember their signs (- for sine).
- Forgetting to include the constant (c) when integrating.
- Misapplying the general result for linear integrals by forgetting the need for the derivative adjustment (\frac{1}{a}).
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