Determine The Following Indefinite Integral. Check Your Work By Differentiation.

Finding indefinite integrals is a fundamental skill in calculus. The process, often called antidifferentiation, involves determining a function whose derivative is equal to the given integrand. Once you’ve found a potential solution, it’s crucial to verify your work through differentiation, ensuring the derivative of your result matches the original integrand. This article provides a comprehensive guide on how to determine indefinite integrals and check your solutions, complete with examples and explanations.

Understanding Indefinite Integrals

An indefinite integral represents the family of all functions that have the same derivative. It is denoted by the integral symbol ∫, followed by the integrand (the function to be integrated) and the differential dx. The result of an indefinite integral is a function, F(x), plus a constant of integration, C. This constant is essential because the derivative of a constant is always zero, meaning there are infinitely many functions that could have the same derivative.

Mathematically, we can express this as:

∫ f(x) dx = F(x) + C

Where:

• ∫ is the integral symbol
• f(x) is the integrand
• dx indicates that the integration is performed with respect to x
• F(x) is the antiderivative of f(x)
• C is the constant of integration

Basic Integration Rules

To effectively determine indefinite integrals, you need to be familiar with several fundamental integration rules. Here are some of the most commonly used rules:

1. Power Rule:

   ∫ xⁿ dx = (xⁿ⁺¹ / n+1) + C, where n ≠ -1

   • This rule applies to integrating power functions. Increase the exponent by 1 and divide by the new exponent.

2. Constant Multiple Rule:

   ∫ k f(x) dx = k ∫ f(x) dx, where k is a constant

   • You can move a constant factor outside the integral.

3. Sum/Difference Rule:

   ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

   • The integral of a sum or difference of functions is the sum or difference of their individual integrals.

4. Integral of 1:

   ∫ 1 dx = x + C

5. Integral of a Constant:

   ∫ k dx = kx + C, where k is a constant

6. Exponential Rule:

   ∫ e^x dx = e^x + C ∫ a^x dx = (a^x / ln a) + C, where a > 0 and a ≠ 1

7. Trigonometric Integrals:

   • ∫ sin x dx = -cos x + C
   • ∫ cos x dx = sin x + C
   • ∫ sec² x dx = tan x + C
   • ∫ csc² x dx = -cot x + C
   • ∫ sec x tan x dx = sec x + C
   • ∫ csc x cot x dx = -csc x + C

8. Reciprocal Rule:

   ∫ (1/x) dx = ln |x| + C

   • The integral of 1/x is the natural logarithm of the absolute value of x.

Step-by-Step Guide to Finding Indefinite Integrals

Here’s a detailed process to determine indefinite integrals, along with examples to illustrate each step:

Step 1: Simplify the Integrand

Before applying any integration rules, simplify the integrand as much as possible. This might involve expanding expressions, combining like terms, or using trigonometric identities.

Example 1: Find ∫ (3x² + 4x - 5) dx

• The integrand is already in a simplified form, so we can proceed to the next step.

Step 2: Apply Integration Rules

Use the appropriate integration rules to find the antiderivative of each term in the integrand. Remember to include the constant of integration, C, at the end.

Example 1 (Continued):

∫ (3x² + 4x - 5) dx = ∫ 3x² dx + ∫ 4x dx - ∫ 5 dx

Apply the power rule and constant multiple rule:

= 3 ∫ x² dx + 4 ∫ x dx - 5 ∫ 1 dx

= 3 (x³/3) + 4 (x²/2) - 5x + C

Simplify:

= x³ + 2x² - 5x + C

Step 3: Check Your Work by Differentiation

To verify your solution, differentiate the antiderivative you found in Step 2. The derivative should be equal to the original integrand.

Example 1 (Continued):

Differentiate x³ + 2x² - 5x + C with respect to x:

d/dx (x³ + 2x² - 5x + C) = 3x² + 4x - 5

Since the derivative matches the original integrand (3x² + 4x - 5), our solution is correct.

Examples of Indefinite Integrals with Verification

Let's explore more examples to solidify your understanding:

Example 2: Find ∫ (5e^x - 2/ x) dx

Step 1: Simplify the Integrand

The integrand is already simplified.

Step 2: Apply Integration Rules

∫ (5e^x - 2/ x) dx = 5 ∫ e^x dx - 2 ∫ (1/x) dx

Apply the exponential and reciprocal rules:

= 5e^x - 2 ln |x| + C

Step 3: Check Your Work by Differentiation

Differentiate 5e^x - 2 ln |x| + C with respect to x:

d/dx (5e^x - 2 ln |x| + C) = 5e^x - 2/x

The derivative matches the original integrand, so our solution is correct.

Example 3: Find ∫ (sin x + cos x) dx

Step 1: Simplify the Integrand

The integrand is already simplified.

Step 2: Apply Integration Rules

∫ (sin x + cos x) dx = ∫ sin x dx + ∫ cos x dx

Apply the trigonometric integration rules:

= -cos x + sin x + C

Step 3: Check Your Work by Differentiation

Differentiate -cos x + sin x + C with respect to x:

d/dx (-cos x + sin x + C) = sin x + cos x

The derivative matches the original integrand, confirming our solution.

Example 4: Find ∫ ( x³ + 2 )² dx

Step 1: Simplify the Integrand

Expand the expression:

( x³ + 2 )² = x⁶ + 4x³ + 4

So, we need to find ∫ (x⁶ + 4x³ + 4) dx

Step 2: Apply Integration Rules

∫ (x⁶ + 4x³ + 4) dx = ∫ x⁶ dx + 4 ∫ x³ dx + 4 ∫ 1 dx

Apply the power rule:

= (x⁷/7) + 4(x⁴/4) + 4x + C

Simplify:

= (x⁷/7) + x⁴ + 4x + C

Step 3: Check Your Work by Differentiation

Differentiate (x⁷/7) + x⁴ + 4x + C with respect to x:

d/dx [(x⁷/7) + x⁴ + 4x + C] = x⁶ + 4x³ + 4

Since x⁶ + 4x³ + 4 = ( x³ + 2 )², the derivative matches the original integrand, so our solution is correct.

Techniques for More Complex Integrals

While basic integration rules cover many common functions, more complex integrals may require advanced techniques such as:

• U-Substitution: This technique is used when the integrand can be expressed in the form f(g(x)) g'(x). By substituting u = g(x), the integral can be simplified.
• Integration by Parts: This technique is used to integrate the product of two functions. The formula is ∫ u dv = uv - ∫ v du.
• Trigonometric Substitution: This technique is used when the integrand contains expressions of the form √(a² - x²), √(a² + x²), or √(x² - a²).
• Partial Fraction Decomposition: This technique is used to integrate rational functions (ratios of polynomials).

These advanced techniques enable you to tackle a wider range of integrals, but they also require a deeper understanding of calculus principles.

Common Mistakes to Avoid

When determining indefinite integrals, it's important to be aware of common mistakes to avoid:

1. Forgetting the Constant of Integration: Always remember to add the constant of integration, C, to the end of every indefinite integral.
2. Incorrectly Applying the Power Rule: Ensure that you correctly apply the power rule, especially when dealing with negative or fractional exponents. Remember that the power rule does not apply when n = -1.
3. Not Simplifying the Integrand: Always simplify the integrand as much as possible before applying integration rules. This can make the integration process much easier.
4. Making Errors in Differentiation: When checking your work by differentiation, be careful to avoid errors in applying differentiation rules. Double-check your work to ensure that the derivative matches the original integrand.
5. Incorrectly Applying U-Substitution: When using u-substitution, ensure that you correctly identify the u and du, and that you properly substitute them into the integral.

By being mindful of these common mistakes, you can improve your accuracy and confidence in determining indefinite integrals.

Examples using U-Substitution

Example 5: Find ∫ 2x cos(x²) dx

Step 1: Identify u and du

Let u = x²

Then, du = 2x dx

Step 2: Substitute into the Integral

∫ 2x cos(x²) dx = ∫ cos(u) du

Step 3: Integrate

∫ cos(u) du = sin(u) + C

Step 4: Substitute back

sin(u) + C = sin(x²) + C

Step 5: Check Your Work by Differentiation

Differentiate sin(x²) + C with respect to x:

d/dx [sin(x²) + C] = cos(x²) * 2x = 2x cos(x²)

The derivative matches the original integrand, so our solution is correct.

Example 6: Find ∫ x √( x + 1 ) dx

Step 1: Identify u and du

Let u = x + 1

Then, du = dx

Also, x = u - 1

Step 2: Substitute into the Integral

∫ x √( x + 1 ) dx = ∫ (u - 1) √u du = ∫ (u - 1) u^1/2 du

Step 3: Simplify and Integrate

∫ (u^3/2 - u^1/2) du = ∫ u^3/2 du - ∫ u^1/2 du

= (u^5/2/(5/2)) - (u^3/2/(3/2)) + C

= (2/5) u^5/2 - (2/3) u^3/2 + C

Step 4: Substitute back

= (2/5) (x + 1)^5/2 - (2/3) (x + 1)^3/2 + C

Step 5: Check Your Work by Differentiation

Differentiate (2/5) (x + 1)^5/2 - (2/3) (x + 1)^3/2 + C with respect to x:

d/dx [(2/5) (x + 1)^5/2 - (2/3) (x + 1)^3/2 + C] = (x + 1)^3/2 - (x + 1)^1/2 = (x + 1)^1/2 [ (x + 1) - 1 ] = x √( x + 1 )

The derivative matches the original integrand, so our solution is correct.

Conclusion

Determining indefinite integrals is a core skill in calculus that requires a solid understanding of basic integration rules and techniques. By following a systematic approach, simplifying the integrand, applying the appropriate rules, and always checking your work by differentiation, you can confidently find indefinite integrals and ensure the accuracy of your solutions. As you gain more experience, you can explore more advanced techniques to tackle complex integrals. Remember, practice is key to mastering this fundamental aspect of calculus.