Solved Exercises on Standard Forms of Integration in Grade 1

Master standard forms of integration: power rule, exponential, logarithmic, and trigonometric integrals through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Power rule integration
Exercise 1
Evaluate the integral:
\(\int x^3 \, dx\)
Definition:

Power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where \(n \neq -1\)

Integration method:
  1. Identify the power n
  2. Add 1 to the exponent
  3. Divide by the new exponent
  4. Add constant of integration C
Integral
\(\int x^3 \, dx\)
Apply power rule
\(\frac{x^{3+1}}{3+1}\)
Result
\(\frac{x^4}{4} + C\)
Step 1: Identify the power

Here n = 3 in the function x³

Step 2: Apply the power rule

Formula: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

\(\int x^3 \, dx = \frac{x^{3+1}}{3+1} + C\)

Step 3: Simplify the expression

\(\frac{x^{3+1}}{3+1} = \frac{x^4}{4}\)

Step 4: Add constant of integration

\(\int x^3 \, dx = \frac{x^4}{4} + C\)

\(\int x^3 \, dx = \frac{x^4}{4} + C\)
Final answer:

\(\int x^3 \, dx = \frac{x^4}{4} + C\)

Applied rules:

• Power rule: Increase exponent by 1, divide by new exponent

• Integration constant: Always add C for indefinite integrals

• Constraint: Power rule doesn't apply when n = -1

2 Exponential integration
Exercise 2
Evaluate the integral:
\(\int e^{2x} \, dx\)
Definition:

Exponential rule: \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\) where a ≠ 0

Integral
\(\int e^{2x} \, dx\)
Apply rule
\(\frac{1}{2}e^{2x}\)
Result
\(\frac{1}{2}e^{2x} + C\)
Step 1: Identify the coefficient

Here a = 2 in the function e^(2x)

Step 2: Apply the exponential rule

Formula: \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\)

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

Step 3: Simplify the expression

The coefficient of the exponential function is the reciprocal of the exponent's coefficient

Step 4: Add constant of integration

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

\(\int e^{2x} \, dx = \frac{1}{2}e^{2x} + C\)
Final answer:

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

Applied rules:

• Exponential rule: Integral of e^(ax) is (1/a)e^(ax)

• Chain rule reversal: Division by coefficient compensates for derivative of inner function

• Integration constant: Always add C for indefinite integrals

3 Logarithmic integration
Exercise 3
Evaluate the integral:
\(\int \frac{1}{x} \, dx\)
Definition:

Logarithmic rule: \(\int \frac{1}{x} \, dx = \ln|x| + C\)

Integral
\(\int \frac{1}{x} \, dx\)
Apply rule
\(\ln|x|\)
Result
\(\ln|x| + C\)
Step 1: Recognize the special case

This is the exception to the power rule where n = -1

Step 2: Apply the logarithmic rule

Formula: \(\int \frac{1}{x} \, dx = \ln|x| + C\)

Step 3: Note the absolute value

The absolute value ensures the function is defined for both positive and negative x

Step 4: Add constant of integration

\(\int \frac{1}{x} \, dx = \ln|x| + C\)

\(\int \frac{1}{x} \, dx = \ln|x| + C\)
Final answer:

\(\int \frac{1}{x} \, dx = \ln|x| + C\)

Applied rules:

• Logarithmic rule: Special case for x^(-1)

• Absolute value: Ensures domain includes negative values

• Integration constant: Always add C for indefinite integrals

Integration Rules and Methods
\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)
Power Rule
Power Rule
\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)
n ≠ -1
Exponential
\(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\)
a ≠ 0
Logarithmic
\(\int \frac{1}{x} \, dx = \ln|x| + C\)
Special case
Key definitions:

Integration: Process of finding the antiderivative of a function

Antiderivative: Function whose derivative is the original function

Constant of integration: Arbitrary constant C added to indefinite integrals

Indefinite integral: Integral without limits, representing a family of functions

Complete methodology:
  1. Analyze the function: Identify the type of function (power, exponential, trigonometric, etc.)
  2. Select the appropriate rule: Match the function to known integration formulas
  3. Apply the rule: Follow the specific procedure for the identified function type
  4. Verify the result: Differentiate to check if you get the original function
Tip 1: Always add the constant of integration C for indefinite integrals.
Tip 2: Remember that the power rule doesn't work when n = -1.
Tip 3: For exponential functions, divide by the coefficient of x in the exponent.
Tip 4: Use absolute value for ln|x| to ensure the function is defined for negative x.
Common errors: Forgetting the constant C, applying power rule when n = -1, missing absolute value in logarithm.
Exam preparation: Memorize basic integration formulas, practice pattern recognition, understand the relationship between differentiation and integration.
Formulas to know by heart:

• Power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) (n ≠ -1)

• Exponential: \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\) (a ≠ 0)

• Logarithmic: \(\int \frac{1}{x} \, dx = \ln|x| + C\)

• Trigonometric: \(\int \sin x \, dx = -\cos x + C\), \(\int \cos x \, dx = \sin x + C\)

• Constant: \(\int a \, dx = ax + C\)

Solution: Exercises 4 to 5
4 Trigonometric integration
Exercise 4
Evaluate the integral:
\(\int \cos(3x) \, dx\)
Definition:

Trigonometric rule: \(\int \cos(ax) \, dx = \frac{1}{a}\sin(ax) + C\)

Integral
\(\int \cos(3x) \, dx\)
Apply rule
\(\frac{1}{3}\sin(3x)\)
Result
\(\frac{1}{3}\sin(3x) + C\)
Step 1: Identify the coefficient

Here a = 3 in the function cos(3x)

Step 2: Apply the cosine rule

Formula: \(\int \cos(ax) \, dx = \frac{1}{a}\sin(ax) + C\)

\(\int \cos(3x) \, dx = \frac{1}{3}\sin(3x) + C\)

Step 3: Verify the result

Differentiate: \(\frac{d}{dx}[\frac{1}{3}\sin(3x)] = \frac{1}{3} \cdot 3\cos(3x) = \cos(3x)\) ✓

Step 4: Add constant of integration

\(\int \cos(3x) \, dx = \frac{1}{3}\sin(3x) + C\)

\(\int \cos(3x) \, dx = \frac{1}{3}\sin(3x) + C\)
Final answer:

\(\int \cos(3x) \, dx = \frac{1}{3}\sin(3x) + C\)

Applied rules:

• Cosine rule: Integral of cos(ax) is (1/a)sin(ax)

• Chain rule reversal: Division by coefficient compensates for derivative of inner function

• Integration constant: Always add C for indefinite integrals

5 Mixed integration
Exercise 5
Evaluate the integral:
\(\int (4x^3 + 2e^x - \frac{5}{x}) \, dx\)
Definition:

Linearity of integration: \(\int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx\)

Original
\(\int (4x^3 + 2e^x - \frac{5}{x}) \, dx\)
Separate
\(4\int x^3 \, dx + 2\int e^x \, dx - 5\int \frac{1}{x} \, dx\)
Result
\(x^4 + 2e^x - 5\ln|x| + C\)
Step 1: Apply linearity property

Split the integral: \(\int (4x^3 + 2e^x - 5x^{-1}) \, dx = 4\int x^3 \, dx + 2\int e^x \, dx - 5\int \frac{1}{x} \, dx\)

Step 2: Integrate each term separately

Term 1: \(4\int x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4\)

Term 2: \(2\int e^x \, dx = 2e^x\)

Term 3: \(-5\int \frac{1}{x} \, dx = -5\ln|x|\)

Step 3: Combine all terms

\(x^4 + 2e^x - 5\ln|x|\)

Step 4: Add constant of integration

\(\int (4x^3 + 2e^x - \frac{5}{x}) \, dx = x^4 + 2e^x - 5\ln|x| + C\)

\(\int (4x^3 + 2e^x - \frac{5}{x}) \, dx = x^4 + 2e^x - 5\ln|x| + C\)
Final answer:

\(\int (4x^3 + 2e^x - \frac{5}{x}) \, dx = x^4 + 2e^x - 5\ln|x| + C\)

Applied rules:

• Linearity: Split complex integrals into simpler parts

• Constant multiple: Pull constants outside the integral

• Combination: Apply appropriate rule to each term individually

Integration Summary and Key Concepts
\(\int f(x) \, dx = F(x) + C\)
Fundamental Integration Formula
Key definitions:

Integration: The reverse process of differentiation, also called antidifferentiation

Antiderivative: A function F(x) such that F'(x) = f(x)

Indefinite integral: The family of all antiderivatives of f(x), denoted ∫f(x)dx

Constant of integration: The arbitrary constant C representing the family of functions

Definite integral: An integral with specified upper and lower limits

Complete methodology:
  1. Pattern recognition: Identify the form of the integrand
  2. Rule selection: Choose appropriate integration technique
  3. Application: Apply the selected rule systematically
  4. Verification: Differentiate the result to confirm accuracy
Tip 1: Always check your answer by differentiating the result.
Tip 2: Remember that integration is the inverse operation of differentiation.
Tip 3: For exponential functions with base e, the integral is straightforward.
Tip 4: When integrating sums, integrate each term separately using linearity.
Integration applications: Finding areas under curves, solving differential equations, calculating volumes.
Key relationships: \(\frac{d}{dx}[∫f(x)dx] = f(x)\), \(∫[\frac{d}{dx}F(x)]dx = F(x) + C\).
Essential formulas:

• Power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) (n ≠ -1)

• Exponential: \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\) (a ≠ 0)

• Logarithmic: \(\int \frac{1}{x} \, dx = \ln|x| + C\)

• Trigonometric: \(\int \sin x \, dx = -\cos x + C\), \(\int \cos x \, dx = \sin x + C\)

• Linearity: \(\int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx\)

Exercise with Visualization: Integration Families
Exercise 6: Integration of Function Families
Analyze how integration affects different function families:
\(F_1(x) = \int x^2 \, dx = \frac{x^3}{3}\)
\(F_2(x) = \int e^x \, dx = e^x\)
\(F_3(x) = \int \sin x \, dx = -\cos x\)

Analysis: The visualization shows how integration transforms different function types.

  • \(F_1(x) = \frac{x^3}{3}\): Cubic function from quadratic input
  • \(F_2(x) = e^x\): Exponential function preserves its form
  • \(F_3(x) = -\cos x\): Trigonometric transformation

Questions & Answers

Question: I don't understand why we add a constant C when integrating. Why isn't there a constant when we differentiate?

Answer: This is a fundamental concept! When we differentiate a function, any constant term disappears because the derivative of a constant is zero. For example:

  • \(\frac{d}{dx}[x^2 + 5] = 2x\)
  • \(\frac{d}{dx}[x^2 + 10] = 2x\)
  • \(\frac{d}{dx}[x^2 + C] = 2x\) for any constant C

This means that when we integrate (reverse the differentiation), we don't know what the original constant was. The function x² could have come from differentiating x² + 5, x² + 10, or x² + C for any value of C.

Therefore, when we write \(\int 2x \, dx = x^2 + C\), the "+C" represents the unknown constant that could have been part of the original function. This creates a "family" of functions that all have the same derivative.

Question: Why doesn't the power rule work when n = -1? What makes x⁻¹ special?

Answer: Let's see what happens when we try to apply the power rule to x⁻¹:

If we naively applied the power rule: \(\int x^{-1} \, dx = \frac{x^{-1+1}}{-1+1} = \frac{x^0}{0} = \frac{1}{0}\)

Division by zero is undefined! This is why the power rule breaks down for n = -1.

The function f(x) = 1/x is special because its antiderivative is the natural logarithm function. This can be proven using the definition of the natural logarithm as an area under the curve y = 1/t from 1 to x.

So we have a special rule: \(\int \frac{1}{x} \, dx = \ln|x| + C\)

Note the absolute value: ln|x| works for both positive and negative values of x, since 1/x is defined for all x ≠ 0.

Question: When integrating functions like e^(2x), why do we divide by the coefficient of x in the exponent?

Answer: This comes from the chain rule of differentiation. Let's look at what happens when we differentiate e^(2x):

  • \(\frac{d}{dx}[e^{2x}] = e^{2x} \cdot \frac{d}{dx}[2x] = e^{2x} \cdot 2 = 2e^{2x}\)

So the derivative of e^(2x) is 2e^(2x), not just e^(2x).

When we integrate, we're reversing this process. We want to find a function whose derivative is e^(2x). Since we know that the derivative of e^(2x) brings down a factor of 2, we need to compensate by dividing by 2:

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

Let's verify: \(\frac{d}{dx}[\frac{1}{2}e^{2x}] = \frac{1}{2} \cdot 2e^{2x} = e^{2x}\) ✓

This is a general principle: when integrating e^(ax), divide by the coefficient a to compensate for the chain rule effect.

Question: How can I remember all these integration formulas? There seem to be too many to memorize.

Answer: Rather than rote memorization, focus on understanding the relationships:

  1. Connect to derivatives: Remember that integration is the reverse of differentiation
  2. Understand patterns: Notice that many rules involve dividing by coefficients (chain rule reversal)
  3. Focus on fundamentals: Master the power rule, exponential rule, and logarithmic rule first

Key memory aids:

  • Power rule: Add 1 to exponent, divide by new exponent
  • Exponential: Keep the exponential, divide by coefficient of x in exponent
  • 1/x: Always gives ln|x| (special case)

Practice regularly with verification: after integrating, always differentiate your answer to check if you get back to the original function. This reinforces the connection between integration and differentiation.

The more you practice, the more these patterns become intuitive rather than memorized.

Question: Are there any real-world applications of integration that I can relate to?

Answer: Integration has numerous practical applications:

  1. Area calculation: Finding the area under a curve, like the area under a velocity-time graph to get displacement
  2. Volume calculation: Finding volumes of irregular shapes by slicing them into thin pieces
  3. Physics applications: Calculating work done, center of mass, moments of inertia
  4. Economics: Calculating consumer surplus, total cost from marginal cost
  5. Biology: Modeling population growth, drug concentration over time

Simple example: If you know how fast a car is going at every moment (velocity), integration tells you how far it traveled (distance).

More complex example: If you know the rate of water flowing into a tank, integration tells you the total amount of water accumulated.

Integration essentially accumulates quantities that change continuously, making it invaluable in science, engineering, economics, and many other fields!