Solving Calculus Problems On Integration

 

Mastering Integration

Definition, 10 Core Rules, Practice Questions, and Detailed Solutions

What is Integration?

Integration is a foundational concept in calculus that serves as the inverse operation to differentiation. While differentiation determines the rate of change at a given point, integration accumulates these values to calculate total sizes, most notably the net area bounded under a curve.

• Indefinite Integrals: Represent families of functions and always include an arbitrary constant (C).
• Definite Integrals: Evaluated across specific physical intervals to provide a concrete numeric value.

1. The Power Rule

Used to integrate variable bases raised to fixed numerical exponents.

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

Question 1.1: Find the indefinite integral: ∫ x⁴ dx.

• A) 4x³ + C
• B) (1/5)x⁵ + C
• C) 5x⁵ + C
• D) x⁵ + C

Correct Answer: B

Solution: Raise power by 1 and divide by the new exponent: (4+1=5). Result = (1/5)x⁵ + C.

Question 1.2: Evaluate: ∫ 1/x³ dx.

• A) -1/(2x²) + C
• B) 1/(4x⁴) + C
• C) -3/x⁴ + C
• D) log(x³) + C

Correct Answer: A

Solution: Rewrite as negative exponent: ∫ x^-3 dx. Apply rule: x^-2 / -2 + C = -1/(2x²) + C.

2. Constant & Constant Multiple Rule

Constants can be factored directly out of integration operations cleanly.

∫ k dx = kx + C  |  ∫ k·f(x) dx = k ∫ f(x) dx

Question 2.1: Find ∫ 7 dx.

• A) 7 + C
• B) 0 + C
• C) 7x + C
• D) 7/2 x² + C

Correct Answer: C

Solution: The integral of any standalone constant k with respect to x is always kx + C. Thus, 7x + C.

Question 2.2: Evaluate ∫ 6x² dx.

• A) 2x³ + C
• B) 18x³ + C
• C) 6x³ + C
• D) 3x² + C

Correct Answer: A

Solution: Factor out 6: 6 ∫ x² dx = 6 · (x³ / 3) = 2x³ + C.

3. Sum and Difference Rule

Integrals can be distributed cleanly across independent additive or subtractive elements.

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

Question 3.1: Evaluate ∫ (3x² + 2x) dx.

• A) 6x + 2 + C
• B) x³ + x² + C
• C) 3x³ + 2x² + C
• D) x³ + 2x + C

Correct Answer: B

Solution: Split terms: ∫ 3x² dx + ∫ 2x dx = x³ + x² + C. Result = x³ + x² + C.

Question 3.2: Find ∫ (5 - x) dx.

• A) 5x - (1/2)x² + C
• B) 5x - x + C
• C) -1 + C
• D) 5x - x² + C

Correct Answer: A

Solution: Integrating independent blocks gives: ∫ 5 dx - ∫ x dx = 5x - (1/2)x² + C.

4. The Exponential Rule

Natural exponential operations are unique as they serve as their own primary integrals.

∫ e^x dx = e^x + C  |  ∫ e^ax dx = (1/a)e^ax + C

Question 4.1: Evaluate ∫ 4e^x dx.

• A) e^x + C
• B) 4e^x + C
• C) 2e^2x + C
• D) 4xe^x-1 + C

Correct Answer: B

Solution: Pull out the constant scale factor directly: 4 ∫ e^x dx = 4e^x + C.

Question 4.2: Find ∫ e^5x dx.

• A) 5e^5x + C
• B) e^5x + C
• C) (1/5)e^5x + C
• D) e^6x/6 + C

Correct Answer: C

Solution: Divide structural expression by the internal derivative coefficient (5): (1/5)e^5x + C.

5. The Reciprocal Rule (Logarithmic Rule)

The solution when evaluating functions with a power profile of exactly -1.

∫ (1/x) dx = ln|x| + C

Question 5.1: Find the integral: ∫ 3/x dx.

• A) -3/x² + C
• B) 3 ln|x| + C
• C) ln|3x| + C
• D) 3x⁰ + C

Correct Answer: B

Solution: Separate the numerator value: 3 ∫ (1/x) dx = 3 ln|x| + C.

Question 5.2: Evaluate ∫ 1/(4x) dx.

• A) (1/4) ln|x| + C
• B) ln|4x| + C
• C) 4 ln|x| + C
• D) -1/(16x²) + C

Correct Answer: A

Solution: Extract scalar fractions cleanly before processing: (1/4) ∫ (1/x) dx = (1/4) ln|x| + C.

6. Trigonometric Rules (Sine and Cosine)

Core cyclic rules tracking foundational derivatives in inverse format.

∫ sin(x) dx = -cos(x) + C  |  ∫ cos(x) dx = sin(x) + C

Question 6.1: Evaluate ∫ -2 cos(x) dx.

• A) 2 sin(x) + C
• B) -2 sin(x) + C
• C) -2 cos(x) + C
• D) 2 cos(x) + C

Correct Answer: B

Solution: The integral of cos(x) is sin(x). Multiplying by -2 yields -2 sin(x) + C.

Question 6.2: Find ∫ (x - sin(x)) dx.

• A) (1/2)x² - cos(x) + C
• B) x² + cos(x) + C
• C) (1/2)x² + cos(x) + C
• D) 1 - cos(x) + C

Correct Answer: C

Solution: ∫ x dx - ∫ sin(x) dx = (1/2)x² - (-cos(x)) + C = (1/2)x² + cos(x) + C.

7. Advanced Trigonometric Rules (Secant Squared)

Since the derivative of tangent functions equals secant squared, integration reverses this track perfectly.

∫ sec²(x) dx = tan(x) + C

Question 7.1: Evaluate ∫ 5 sec²(x) dx.

• A) 5 tan(x) + C
• B) -5 tan(x) + C
• C) 5 sec(x) + C
• D) 5/3 sec³(x) + C

Correct Answer: A

Solution: Factor out the scale parameter: 5 ∫ sec²(x) dx = 5 tan(x) + C.

Question 7.2: Find ∫ sec²(2x) dx.

• A) tan(2x) + C
• B) 2 tan(2x) + C
• C) (1/2) tan(2x) + C
• D) sec(2x) tan(2x) + C

Correct Answer: C

Solution: Divide by the internal linear derivative coefficient (2) to scale back the transformation: (1/2) tan(2x) + C.

8. Integration by Substitution (u-Substitution)

The inverse version of the Chain Rule. Used when an expression contains both an inner function and its derivative.

∫ f(g(x))·g'(x) dx = ∫ f(u) du, where u = g(x)

Question 8.1: Evaluate ∫ 2x e^x2 dx.

• A) e^x2 + C
• B) 2e^x2 + C
• C) x²e^x + C
• D) e^x + C

Correct Answer: A

Solution: Set u = x², giving du = 2x dx. Substituting gives ∫ e^u du = e^u + C = e^x2 + C.

Question 8.2: Find ∫ (2x + 3)⁵ dx.

• A) (2x + 3)⁶ / 6 + C
• B) (2x + 3)⁶ / 12 + C
• C) 2(2x + 3)⁶ + C