DerivativeCalculus.com

Integration Mastery

Master the art of integration with 15 comprehensive problems covering basic to advanced techniques, including u-substitution and applications.

📚 Fundamental Integration Rules
∫ xⁿ dx
= (xⁿ⁺¹)/(n+1) + C, n ≠ -1
∫ eˣ dx
= eˣ + C
∫ 1/x dx
= ln|x| + C
∫ sin(x) dx
= -cos(x) + C
∫ cos(x) dx
= sin(x) + C
∫ sec²(x) dx
= tan(x) + C
Power Rule for Integration
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C

Increase exponent by 1, divide by new exponent

Constant Multiple Rule
∫ k·f(x) dx = k·∫ f(x) dx

Constants can be factored out

Sum/Difference Rule
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Integrate term by term

Key Insight: Integration is the reverse of differentiation. The +C (constant of integration) is crucial for indefinite integrals because many functions have the same derivative.

Problem 1: Basic Indefinite Integral Beginner
∫ (3x² - 4x + 5) dx
Find the indefinite integral. Don't forget the constant of integration (+C).
1 Apply the sum rule and integrate term by term:
∫ (3x² - 4x + 5) dx = ∫ 3x² dx - ∫ 4x dx + ∫ 5 dx
2 Apply power rule to each term:
∫ 3x² dx = 3·(x³/3) = x³
∫ 4x dx = 4·(x²/2) = 2x²
∫ 5 dx = 5x
3 Combine results and add constant:
x³ - 2x² + 5x + C
x³ - 2x² + 5x + C
Verification

Differentiate your answer to check:

d/dx [x³ - 2x² + 5x + C] = 3x² - 4x + 5 ✓
Problem 2: Definite Integral Intermediate
∫₀² (x³) dx
Evaluate the definite integral from x=0 to x=2.
1 Find the antiderivative:
F(x) = ∫ x³ dx = (1/4)x⁴
2 Apply the Fundamental Theorem of Calculus:
∫₀² x³ dx = F(2) - F(0)
3 Evaluate at upper limit:
F(2) = (1/4)(2⁴) = (1/4)(16) = 4
4 Evaluate at lower limit:
F(0) = (1/4)(0⁴) = 0
5 Calculate the difference:
F(2) - F(0) = 4 - 0 = 4
4
Interpretation

The definite integral ∫₀² x³ dx = 4 represents the area under the curve y = x³ from x=0 to x=2.

Problem 3: Trigonometric Integral Intermediate
∫ (3sin(x) - 2cos(x)) dx
Find the indefinite integral.
1 Integrate term by term:
∫ (3sin(x) - 2cos(x)) dx = 3∫ sin(x) dx - 2∫ cos(x) dx
2 Recall basic trig integrals:
∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
3 Apply to each term:
3∫ sin(x) dx = 3(-cos(x)) = -3cos(x)
2∫ cos(x) dx = 2(sin(x)) = 2sin(x)
4 Combine and add constant:
-3cos(x) - 2sin(x) + C
-3cos(x) - 2sin(x) + C
Common Mistake Alert

Remember: ∫ sin(x) dx = -cos(x) + C, NOT cos(x) + C

∫ cos(x) dx = sin(x) + C, NOT -sin(x) + C

Problem 4: U-Substitution Advanced
∫ 2x·e^(x²) dx
Use u-substitution to find the indefinite integral.
1 Let u = x² (the exponent's inside function):
u = x²
du/dx = 2x ⇒ du = 2x dx
2 Substitute into the integral:
∫ 2x·e^(x²) dx = ∫ e^u du
3 Integrate with respect to u:
∫ e^u du = e^u + C
4 Substitute back u = x²:
e^u + C = e^(x²) + C
e^(x²) + C
U-Substitution Strategy
  1. Choose u = inner function
  2. Find du/dx and solve for du
  3. Substitute u and du into integral
  4. Integrate with respect to u
  5. Substitute back u = original variable
Problem 5: Area Between Curves Advanced
Find area between y = x² and y = 2x from x = 0 to x = 2
Set up and evaluate the definite integral for the area between these curves.
1 Determine which curve is on top:
At x=1: y = 2x = 2, y = x² = 1
So y = 2x is above y = x² on [0,2]
2 Set up area integral:
Area = ∫₀² [top - bottom] dx = ∫₀² [2x - x²] dx
3 Find antiderivative:
∫ (2x - x²) dx = x² - (1/3)x³
4 Evaluate from 0 to 2:
F(2) = (2)² - (1/3)(2)³ = 4 - 8/3 = 12/3 - 8/3 = 4/3
F(0) = (0)² - (1/3)(0)³ = 0
5 Calculate area:
Area = F(2) - F(0) = 4/3 - 0 = 4/3
Area = 4/3 square units
Area Between Curves Formula
Area = ∫ₐᵇ [f(x) - g(x)] dx

where f(x) is the upper curve and g(x) is the lower curve on [a,b]

🎯 Real-World Applications of Integration
📈 Area Under Curves

Definite integrals calculate area

Area = ∫ₐᵇ f(x) dx

Used in economics for total revenue, physics for work done, and statistics for probability

📊 Accumulation Problems

Integrals measure total accumulation

Total = ∫ rate dt

Examples: total distance from velocity, total population growth from growth rate

🔬 Average Value

Average value of a function

f_avg = (1/(b-a)) ∫ₐᵇ f(x) dx

Used to find average temperature, average speed, or average concentration

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🔧 Essential Integration Techniques
U-Substitution
∫ f(g(x))·g'(x) dx = ∫ f(u) du

Reverse chain rule. Choose u = inner function.

Integration by Parts
∫ u dv = uv - ∫ v du

Derived from product rule. Choose u using LIATE rule.

Trigonometric Integrals
∫ sin²(x) dx = x/2 - sin(2x)/4 + C
∫ cos²(x) dx = x/2 + sin(2x)/4 + C
Common Integration Strategies
  1. Simplify first: Expand, factor, or use trig identities
  2. Look for u-substitution: Chain rule in reverse
  3. Try integration by parts: For products of different function types
  4. Use tables of integrals: For standard forms
  5. Check your answer: Differentiate to verify