Integration Practice with Solutions- Step-by-Step Guide

Integration Practice: What Actually Works

Most students approach integration the wrong way. They memorize formulas, copy steps, and then panic when they see a problem that doesn't match the pattern they memorized. This guide cuts through that noise.

You'll get real practice problems with actual solutions. No theory dumps. No motivational garbage. Just the mechanics you need to actually solve integrals.

Before You Start: The Core Rule

Integration is reverse differentiation. That's it. If you know how to take derivatives, you already know the foundation. The techniques below are just tools to make the process faster.

Every integration problem is asking one question: what function, when differentiated, gives me this?

Essential Integration Formulas

You need these committed to memory. Not "pretty much remembered." Solid.

• ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
• ∫ 1/x dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• ∫ eᵃˣ dx = (eᵃˣ)/a + C

If these aren't automatic, stop here and drill them first. Everything else builds on this.

Integration Practice Problems with Solutions

Problem 1: Basic Power Rule

Solve: ∫ x³ dx

Solution:

Apply the power rule: increase the exponent by 1, divide by the new exponent.

∫ x³ dx = (x⁴)/4 + C

Verify by differentiating: d/dx [x⁴/4] = 4x³/4 = x³ ✓

Problem 2: Coefficient with Power

Solve: ∫ 5x² dx

Solution:

Constants factor out: ∫ 5x² dx = 5 ∫ x² dx

Apply power rule: 5 · (x³/3) + C

Simplify: (5x³)/3 + C

Problem 3: Fractional Exponent

Solve: ∫ √x dx

Solution:

Rewrite √x as x^(1/2)

∫ x^(1/2) dx = (x^(3/2))/(3/2) + C

Multiplying by reciprocal: (2x^(3/2))/3 + C

Or in radical form: (2x√x)/3 + C

Problem 4: Exponential Function

Solve: ∫ 3e²ˣ dx

Solution:

Factor out the constant: 3 ∫ e²ˣ dx

For ∫ eᵃˣ dx, divide by a: ∫ e²ˣ dx = (e²ˣ)/2 + C

Multiply by the outside constant: 3 · (e²ˣ)/2 + C

Result: (3e²ˣ)/2 + C

Problem 5: Trigonometric Function

Solve: ∫ (sin x + cos x) dx

Solution:

Integrate each term separately:

∫ sin x dx = -cos x

∫ cos x dx = sin x

Combine: -cos x + sin x + C

Problem 6: Dividing by x

Solve: ∫ (2x + 3/x) dx

Solution:

Split into separate integrals: ∫ 2x dx + ∫ 3/x dx

First term: 2 · (x²/2) = x²

Second term: 3 · ln|x|

Result: x² + 3ln|x| + C

Integration by Substitution: When Basic Rules Fail

Substitution works when you spot a composite function. The trick is identifying what piece, when differentiated, appears elsewhere in the problem.

Substitution Example

Solve: ∫ 2x · cos(x²) dx

Solution:

Notice x² inside cos. Let u = x².

Then du = 2x dx. This matches the 2x dx in the problem.

Substitute: ∫ cos(u) du = sin(u) + C

Back-substitute: sin(x²) + C

Verify by differentiating: cos(x²) · 2x ✓

Integration by Parts: The Harder Cases

When substitution doesn't work, try parts. The formula:

∫ u dv = uv - ∫ v du

Choose u using LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick whatever comes first in that list.

Parts Example

Solve: ∫ x · eˣ dx

Solution:

Algebraic (x) comes before Exponential (eˣ) in LIATE.

Let u = x, dv = eˣ dx

Then du = dx, v = eˣ

Apply formula: x · eˣ - ∫ eˣ · 1 dx

= x · eˣ - eˣ + C

Factor: eˣ(x - 1) + C

Comparison: When to Use Each Method

Getting Started: Your Practice Routine

Don't try to learn everything at once. Follow this sequence:

1. Day 1-2: Memorize the six basic formulas. Drill them until you can write them without thinking.
2. Day 3-4: Practice power rule problems. Start with integer exponents, then fractions. Do 20 problems minimum.
3. Day 5-6: Add exponential and trig integrals. Mix them with power rule problems.
4. Day 7-8: Learn substitution. Focus on identifying u and du. Do 15 substitution problems.
5. Day 9-10: Tackle integration by parts. Memorize LIATE. Do 10 problems.

Each practice session: no calculator. Write out every step. Check your answer by differentiating it. This is non-negotiable.

Common Mistakes That Kill Your Answers

• Forgetting +C: Every indefinite integral needs the constant. Always.
• Wrong sign on logarithm: ∫ 1/x dx = ln|x|, not ln(x). The absolute value matters.
• Skipping the simplification step: (5x³)/3 is correct. 5x³/3 is correct. 5/3x³ is wrong.
• Substitution errors: If du doesn't match something in the integral, you've picked the wrong u.
• Parts sign errors: The formula is uv - ∫ v du. The minus sign stays. Always.

Quick Reference: Substitution Cheat Sheet

When you see these patterns, try substitution:

• ∫ f(g(x)) · g'(x) dx → let u = g(x)
• ∫ (something) · 2x dx where something contains x²
• ∫ (something) · cos(x) dx where something contains sin(x)
• ∫ (something) · eˣ dx where something contains eˣ

Final Word

Integration skills come from doing problems, not reading about them. This guide gives you the framework. The actual learning happens when you work through 50, 100, 200 problems and stop getting the same types wrong.

Start now. Pick a problem type. Drill it until it's boring. Move to the next. That's the entire method.