Calculos 1 Review Formulas Integration and Diferentiation

This comprehensive guide reviews essential Calculus 1 formulas, focusing on differentiation and integration. We provide clear explanations, practical examples, and an interactive calculator to help you master these fundamental concepts.

Basic Calculus 1 Formulas

Calculus 1 is the foundation of higher mathematics, focusing on two main operations: differentiation and integration. These concepts help analyze functions, rates of change, and accumulation.

Power Rule

If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

For integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (for \( n \neq -1 \)).

Exponential and Logarithmic Functions

Differentiation: \( \frac{d}{dx} e^x = e^x \), \( \frac{d}{dx} \ln x = \frac{1}{x} \).

Integration: \( \int e^x \, dx = e^x + C \), \( \int \frac{1}{x} \, dx = \ln |x| + C \).

Trigonometric Functions

Differentiation: \( \frac{d}{dx} \sin x = \cos x \), \( \frac{d}{dx} \cos x = -\sin x \).

Integration: \( \int \sin x \, dx = -\cos x + C \), \( \int \cos x \, dx = \sin x + C \).

Differentiation Techniques

Differentiation finds the rate at which a function changes. Here are key techniques:

Chain Rule

Used for composite functions: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

Example: If \( y = \sin(3x^2) \), then \( y' = 2x \cdot 3 \cdot \cos(3x^2) = 6x \cos(3x^2) \).

Product Rule

For \( y = u \cdot v \), \( y' = u'v + uv' \).

Quotient Rule

For \( y = \frac{u}{v} \), \( y' = \frac{u'v - uv'}{v^2} \).

Integration Techniques

Integration finds areas under curves and antiderivatives. Key methods include:

Substitution Method

Let \( u = g(x) \), then \( \int f(g(x))g'(x) \, dx = \int f(u) \, du \).

Example: \( \int 2x e^{x^2} \, dx \). Let \( u = x^2 \), then \( du = 2x \, dx \), so \( \int e^u \, du = e^{x^2} + C \).

Integration by Parts

Uses the formula \( \int u \, dv = uv - \int v \, du \).

Partial Fractions

Used to integrate rational functions by breaking them into simpler fractions.

Practical Applications

Calculus 1 has wide applications in physics, engineering, economics, and more:

Finding velocity from position (differentiation)
Calculating areas under curves (integration)
Determining work done by a variable force
Analyzing growth rates in population models

Example: The area under velocity-time graph gives displacement. The integral of \( v(t) = 3t^2 \) from 0 to 2 is \( \int_0^2 3t^2 \, dt = 3 \cdot \frac{t^3}{3} \Big|_0^2 = 4 \).

FAQ

What is the difference between differentiation and integration?

Differentiation finds the rate of change of a function, while integration finds the accumulation of quantities or the area under a curve.

When should I use the chain rule?

Use the chain rule when differentiating a composite function, where one function is nested inside another.

How do I know when to use substitution in integration?

Use substitution when you have a composite function and its derivative appears in the integral.