Integration by Substitution

Integration by Substitution

What is it?

Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.

In simple words

• Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.
• Worked pattern: Integral 2x(x²+1)³ dx=(x²+1)^frac44+C.
• Final check: domain, sign, notation, units, and reasonableness.

Important terms

Integration by substitution: Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.

What you should learn

• State the rule for integration by substitution.
• Solve a integration by substitution problem step by step.
• Check the result, restrictions, notation, and units.

Full school-level explanation

Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.

CONCEPT AND RULE

Choose u for an inner expression whose derivative is also present, change the whole integral, then substitute back.

METHOD

For integration by substitution, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.

WORKED EXAMPLE

Integral 2x(x²+1)³ dx=(x²+1)^frac44+C.

COMMON MISTAKE

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

How to solve integration by substitution

How it works

• Identify the mathematical objects, givens, restrictions, and required result.
• Choose and write the relevant definition, theorem, or formula.
• Substitute with brackets and carry out one justified step per line.
• Simplify, state the final answer clearly, and verify it in the original setting.

Remember

• Integration by substitution
• rule
• worked example
• worked example

How the ideas connect

Worked example: Integral 2x(x²+1)³ dx=(x²+1)^frac44+C.

Deeper understanding

Go beyond memorising Integration by Substitution. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked example: Integral 2x(x²+1)³ dx=(x²+1)^frac44+C.

Common mistakes

Exam focus

Core Idea

Rule / Formula

Method

Worked Example

Exam focus

Quick revision

• Integration by substitution
• rule
• worked example
• Integration by substitution: Choose u for an inner expression whose derivative is also present, change the whole integral, then s
• CONCEPT AND RULE: Choose u for an inner expression whose derivative is also present, change the whole integral, then s
• METHOD: For integration by substitution, identify the given information and required result. Select the matc

Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.

For long answers on Integration by Substitution: define clearly → explain the main parts → give one example → state one exam caution.