Integration Basics

Integration uses the symbol ∫ (an elongated S) and requires "dx" at the end to show we're integrating with respect to x. When we integrate, we're essentially finding the anti-derivative of a function.

The fundamental rule of integration is: ∫ ax^n dx = $ax^(n+1)$/$n+1$ + c

This means you increase the power by one and divide by the new power. The constant of integration (c) must always be included with indefinite integrals to account for any constants that disappear during differentiation.

Pro tip: Always check your integration by differentiating your answer - if you get back to the original expression, you've integrated correctly!

You can think of integration as "undoing" differentiation. Once you master the basic pattern, you'll find it becomes quite straightforward to apply to various expressions.

Multiple Terms and Different Variables

When integrating expressions with multiple terms, you simply integrate each term separately: ∫$f(x) + g(x)$dx = ∫f(x)dx + ∫g(x)dx

For example: ∫$x^4 + 2x^3$dx = $x^5/5$ + $2x^4/4$ + c = $x^5/5$ + $x^4/2$ + c

Integration works with any variable - not just x. The variable after the "d" tells you what you're integrating with respect to. For instance, ∫u^4 du means we're integrating u^4 with respect to u.

Remember: Before integrating, you might need to rewrite expressions using index rules. Convert roots to fractional indices, expand brackets, and rewrite fractions using negative indices.

When preparing expressions for integration, make sure to convert everything to powers of your variable. For example:

• ∫$1/x^4$dx = ∫x^(-4)dx = $x^(-3)/(-3)$ + c = -1/$3x^3$ + c
• ∫$1/√x$dx = ∫x^(-1/2)dx = $x^(1/2)/(1/2)$ + c = 2√x + c

Solving Differential Equations

Differential equations contain derivatives like dy/dx. To solve them, you integrate both sides and then use additional information (typically a point on the curve) to find the constant of integration.

For example, if dy/dx = 8x - 1 and y = 5 when x = 1:

1. Integrate both sides: y = 4x² - x + c
2. Substitute the known values $x=1, y=5$: 5 = 4(1)² - 1 + c
3. Solve for c: c = 2
4. Write the particular solution: y = 4x² - x + 2

This process always follows the same pattern:

1. Integrate the expression for dy/dx
2. Include the constant of integration
3. Substitute the given point to find the value of c
4. Write the final solution

Learning tip: When solving differential equations, think of it as finding which function has the given derivative, then pinpointing exactly which version of that function passes through your specified point.

The method works for any differential equation where you can integrate the right-hand side. Just remember to handle each term carefully during integration.

Definite Integrals

A definite integral evaluates an expression between two specific values (limits) and gives a numerical result. The notation ∫[a to b] f(x)dx represents finding the integral from x=a to x=b.

To evaluate a definite integral:

1. Integrate the expression (without the constant of integration)
2. Substitute the upper limit and calculate the result
3. Substitute the lower limit and calculate the result
4. Subtract: (upper limit result) - (lower limit result)

For example: ∫[1 to 3] x⁴ dx = $x⁵/5$[1 to 3] = (3⁵/5) - (1⁵/5) = 243/5 - 1/5 = 242/5

Definite integrals are powerful tools that allow us to calculate precise values for areas and other quantities.

Watch out: When evaluating definite integrals with negative values or fractional powers, be extra careful with your calculations to avoid sign errors.

The notation [F(x)][a to b] is a shorthand for F(b) - F(a), where F(x) is the integrated expression.

Finding Areas with Integration

Integration allows us to calculate the area between a curve and the x-axis. The formula is: Area = ∫[a to b] f(x)dx

When working with areas:

• Areas above the x-axis give positive values
• Areas below the x-axis give negative values (take the absolute value to find the actual area)
• If a curve crosses the x-axis, you must calculate areas separately above and below, then add their absolute values

For example, to find the area under y = 2x² from x = 0 to x = 4: Area = ∫[0 to 4] 2x² dx = $2x³/3$[0 to 4] = 2(4)³/3 - 0 = 128/3 units²

Visual tip: Sketch the curve whenever possible to see whether areas are above or below the x-axis. This helps you avoid making sign errors in your calculations.

When calculating area between two curves, the formula becomes: Area = ∫[a to b] $upper function - lower function$ dx

The "upper function" is the curve with larger y-values, and the "lower function" is the curve with smaller y-values within the given interval.

Areas Between Curves and Along the y-axis

To find the area between two curves, you need to:

1. Find where the curves intersect (these will be your limits)
2. Determine which curve is the upper and which is the lower
3. Use the formula: Area = ∫[a to b] $upper function - lower function$ dx

For example, the area between y = x² and y = 2x from x = 0 to x = 2: Area = ∫[0 to 2] $2x - x²$ dx = $x² - x³/3$[0 to 2] = (4 - 8/3) - 0 = 4/3 units²

Sometimes it's easier to integrate with respect to y instead of x. In these cases:

1. Rearrange each equation to isolate x
2. Integrate with respect to y: Area = ∫[c to d] $right function - left function$ dy

For instance, for a region bounded by y = ¼x² and x = ±2√y from y = 1 to y = 4: Area = ∫[1 to 4] 2(2√y) dy = ∫[1 to 4] 4y^(1/2) dy = $4(2y^(3/2)/3)$[1 to 4] = 56/3 units²

Problem-solving strategy: When deciding whether to integrate with respect to x or y, choose the approach that gives you simpler expressions to integrate.

Remember that areas are always positive, so the final answer will be in units².