Integration by Substitution Method Explained, Definite integrals, Examples - Calculus

Integration by substitution is a calculus technique to solve integrals of composite functions, acting as the reverse of the chain rule for differentiation. The method involves substituting a part of the integrand with a new variable (often u) to simplify the integral. You identify a function and its derivative within the integrand, set the function as u, then find du (the derivative of u with respect to x multiplied by dx). This transforms the integral into a simpler form, ∫f(u)du, which can be integrated using basic rules. Finally, you substitute the original function back for u to get the antiderivative in terms of the original variable.

💡Steps for Integration by Substitution
• Identify the substitution: Look for an inner function, g(x), whose derivative, g'(x), is also a factor in the integrand.
• Choose u: Let u = g(x).
• Find du: Differentiate u with respect to x to find du/dx, then rearrange to du = g'(x) dx.
• Rewrite the integral: Substitute u and du into the original integral.
• Integrate with respect to u: Solve the new, simpler integral using standard integration rules.
• Substitute back: Replace u with the original function g(x) to get the final answer.

💡Example
Let's integrate ∫ 2x cos(x²) dx:
• Identify: The function cos(x²) has an inner function x², and its derivative is 2x, which is also in the integral.
• Choose u: Let u = x².
• Find du: du/dx = 2x, so du = 2x dx.
• Rewrite: ∫ 2x cos(x²) dx becomes ∫ cos(u) du.
• Integrate: ∫ cos(u) du = sin(u) + C.
• Substitute back: Replace u with x² to get sin(x²) + C.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Integration by substitution
01:12 u-substitution, definite integrals
02:08 Worked example

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