1 · Observe

Our running example:

∫ 2x(x²+1)³ dx

Look at the structure: something is cubed, and sitting next to it is 2x. That 2x is suspicious — it looks like a derivative of x²+1. When the derivative of the inside is hiding in the integrand, substitution will work.

2 · Manipulate

Walk through the solution one step at a time. Each step asks you to predict first — commit to an answer before the reveal.

3 · Predict

Before you start the steps above, commit to these:

4 · See

Live verification: drag x and compare the original integrand with the derivative of our answer (x²+1)⁴/4. If the answer is right, they always match — because differentiating undoes integrating.

5 · Explain

Substitution reverses the chain rule. The chain rule says:

d/dx [ (x²+1)⁴/4 ] = (x²+1)³ · 2x

Reading it backwards: the integrand 2x(x²+1)³ is what you get after the chain rule fires. Setting u = x²+1 collapses the chain back down:

∫ 2x(x²+1)³ dx  →  ∫ u³ du  →  u⁴/4 + C  →  (x²+1)⁴/4 + C

• Step 1: spot the inner function → that’s u.
• Step 2: compute du and hunt for it hiding in the integrand. No hiding du → this u won’t work.
• Step 3: rewrite everything in u (no x’s left!).
• Step 4: integrate in u-land.
• Step 5: substitute back to x-land — and never forget + C.
• Step 6: verify by differentiating.

6 · Challenge

Three rounds. Each round: pick u, then type your final answer (use ^ for powers, e.g. (x^3+5)^3/3). Answers are checked by differentiating them numerically — any equivalent form earns full credit.

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