Integral Examples- Practice Problems and Solutions

What Integrals Actually Are

Integrals are the inverse operation of derivatives. If derivatives measure rates of change, integrals measure accumulation. That's the short version.

You encounter two main types: definite integrals and indefinite integrals. Definite integrals give you a number. Indefinite integrals give you a function plus a constant.

Most students struggle because they try to memorize everything. You don't need that. You need to recognize patterns and apply the right technique.

Indefinite Integrals: The Basics

An indefinite integral looks like this: ∫f(x)dx = F(x) + C

The ∫ symbol means "antiderivative." The dx tells you the variable. The +C accounts for the fact that all antiderivatives differ by a constant.

Power Rule (Your New Best Friend)

The power rule handles anything with x raised to a power:

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

When n = -1, you get ∫(1/x)dx = ln|x| + C

Simple Examples

Example 1: ∫x³ dx

Add 1 to the exponent: 3 + 1 = 4

Divide by the new exponent: x⁴/4 + C

That's it. Done.

Example 2: ∫5x² dx

Constants factor out: 5 ∫x² dx

Apply the rule: 5(x³/3) + C = (5x³)/3 + C

Definite Integrals: Getting Numbers

Definite integrals have bounds. You evaluate the antiderivative at both bounds and subtract.

∫[a to b] f(x) dx = F(b) - F(a)

Example: ∫[0 to 2] 3x² dx

Find the antiderivative: F(x) = 3(x³/3) = x³

Evaluate: F(2) - F(0) = 8 - 0 = 8

The answer is 8. No calculator needed for this one.

Practice Problems

Try these before checking the solutions. Actually solve them. Reading solutions without trying wastes your time.

Problem 1: ∫(4x³ - 2x + 7) dx

Problem 2: ∫(1/x) dx

Problem 3: ∫[1 to 3] (2x + 1) dx

Problem 4: ∫(√x) dx (Hint: √x = x^(1/2))

Solutions

Solution 1:

Integrate each term separately:

∫4x³ dx = 4(x⁴/4) = x⁴
∫(-2x) dx = -2(x²/2) = -x²
∫7 dx = 7x

Combine: x⁴ - x² + 7x + C

Solution 2:

This is the natural logarithm case:

∫(1/x) dx = ln|x| + C

Solution 3:

Antiderivative: F(x) = x² + x

Evaluate: F(3) - F(1) = (9 + 3) - (1 + 1) = 12 - 2 = 10

Solution 4:

Rewrite: x^(1/2)

Apply power rule: (x^(3/2))/(3/2) + C = (2/3)x^(3/2) + C

Common Integration Techniques

The easy integrals end here. Once you hit products, quotients, or compositions, you need techniques.

U-Substitution

U-substitution reverses the chain rule. You substitute to simplify, integrate, then substitute back.

Example: ∫2x(x² + 1)³ dx

Let u = x² + 1

Then du = 2x dx

The integral becomes: ∫u³ du = u⁴/4 + C

Substitute back: (x² + 1)⁴/4 + C

Integration by Parts

For products of different function types. The formula:

∫u dv = uv - ∫v du

Pick u using LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Higher on the list gets u.

Example: ∫x eˣ dx

Let u = x, dv = eˣ dx

Then du = dx, v = eˣ

Apply: xeˣ - ∫eˣ dx = xeˣ - eˣ + C = eˣ(x - 1) + C

Trigonometric Substitution

Use this for expressions with √(a² - x²), √(a² + x²), or √(x² - a²).

The substitution depends on the pattern:

√(a² - x²) → x = a sinθ
√(a² + x²) → x = a tanθ
√(x² - a²) → x = a secθ

Comparison: Integration Techniques

Technique	Use When	Key Step
Power Rule	Single term with x to a power	Add 1 to exponent, divide by new exponent
U-Substitution	Chain rule pattern visible	Substitute u, integrate, back-substitute
Integration by Parts	Product of different types	Choose u using LIATE, apply formula
Trig Substitution	Square roots with quadratic expressions	Match pattern to trig identity, substitute
Partial Fractions	Rational functions with factors in denominator	Decompose into simpler fractions

Getting Started: How to Solve Any Integral

Follow this sequence. Don't skip steps.

Identify the type. Is it a basic power? A product? A quotient? A composition?
Check for substitution opportunities. Look for f(g(x)) · g'(x) patterns. That's your u = g(x).
For products, decide between substitution and integration by parts. If one part is easily differentiable and the other integrable, try parts.
For quotients with polynomials, try partial fractions. Factor the denominator first.
Solve and check. Take the derivative of your answer. You should get back to the original function.

Step 5 catches most mistakes. Do it every time.

Common Mistakes to Avoid

Forgetting the +C on indefinite integrals
Dropping the dx during u-substitution
Wrong choice for u and dv in integration by parts
Not rewriting roots as fractional exponents before applying power rule
Skipping the check by differentiation

These four mistakes account for 90% of wrong answers in homework and exams.

When You're Stuck

If none of these techniques work, the integral might not have an elementary antiderivative. Some functions can't be integrated using basic functions.

∫e^(x²) dx is one example. The answer exists but it can't be written with polynomials, exponentials, logs, or trig functions. In those cases, you either leave it as an integral, approximate numerically, or use special functions.

Your textbook probably won't ask you to integrate something that has no elementary antiderivative unless that's the point of the problem.

Start with the practice problems above. Master the basics. Then move to harder problems. Don't rush.