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📘 Lesson 14 of 20 · Calculus
Integral Calculus

🧰 Integration Techniques

Beyond the basic power rule, several techniques handle more complex integrals — substitution, integration by parts, and partial fractions among the most common.

Course progress: 70%

01 Key Concepts

U-Substitution

Used when an integral contains a function and its derivative. Let u equal the inner function, rewrite in terms of u, integrate, then substitute back.

Integration by Parts

Used for products of functions. Formula: Integral of u dv = uv - Integral of v du. Choose u to be the part that simplifies when differentiated.

Partial Fractions

Used for rational functions; breaks a complex fraction into a sum of simpler fractions that are each easy to integrate.

Trigonometric Integrals

Integrals involving sin, cos, and other trig functions often use identities to simplify before integrating.

Choosing a Technique

Look at the structure: a function times its derivative suggests substitution; a product of unrelated functions suggests integration by parts; a fraction with factorable denominator suggests partial fractions.

02 Key Formulas

03 Solved Examples

Example 1 Find the integral of 2x*(x^2+1)^3 dx using u-substitution.
  1. Let u = x^2+1, so du = 2x dx.
  2. The integral becomes Integral of u^3 du.
  3. Integrate: u^4/4.
  4. Substitute back u = x^2+1.
Answer: (x^2+1)^4 / 4 + C
Example 2 Find the integral of x*e^x dx using integration by parts.
  1. Let u = x (du = dx) and dv = e^x dx (v = e^x).
  2. Apply the formula: uv - Integral of v du = x*e^x - Integral of e^x dx.
  3. Integral of e^x dx = e^x.
  4. Combine: x*e^x - e^x.
Answer: x*e^x - e^x + C
Example 3 Find the integral of cos(x)*sin(x) dx using u-substitution.
  1. Let u = sin(x), so du = cos(x) dx.
  2. The integral becomes Integral of u du.
  3. Integrate: u^2/2.
  4. Substitute back u = sin(x).
Answer: sin^2(x) / 2 + C

04 Practice Questions

1Integrate 3x^2*(x^3+1)^2 dx using substitution (u=x^3+1).
(x^3+1)^3 / 3 + C
2Integrate x*cos(x) dx using integration by parts.
x*sin(x) + cos(x) + C
3Integrate 2x*e^(x^2) dx using substitution.
e^(x^2) + C
4What technique is best for a product of unrelated functions like x and ln(x)?
Integration by parts
5What technique is best when the integral contains a function and its derivative?
U-substitution

📄 Integration Techniques — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet