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Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
Integration
Watch comprehensive video reviews for Integration, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.
Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
Not your average video:
Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.
Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.
Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.
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SL 5.10
By noticing that dxd[f(g(x))+C]=g′(x)⋅f′(g(x)) (chain rule), we can deduce that
This formula is quite obscure, but is made clear by substituting u=g(x)⇒g′(x)=dxdu. The integral then becomes
To find the correct u, we search for one part that is "almost" the derivative of another part. "Almost" because constant multiples k do not fundamentally alter the process.
Example
In the integral∫3x√x2−1dx, we recognize that dxd(x2−1)=2x which is a constant multiple of 3x. So we set u=x2−1, which gives dxdu=2x. Thus the integral becomes:
SL 5.10
By noticing that dxd[f(g(x))+C]=g′(x)⋅f′(g(x)) (chain rule), we can deduce that
This formula is quite obscure, but is made clear by substituting u=g(x)⇒g′(x)=dxdu. The integral then becomes
To find the correct u, we search for one part that is "almost" the derivative of another part. "Almost" because constant multiples k do not fundamentally alter the process.
Example
In the integral∫3x√x2−1dx, we recognize that dxd(x2−1)=2x which is a constant multiple of 3x. So we set u=x2−1, which gives dxdu=2x. Thus the integral becomes: