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Integration techniques

Direct primitives and the chain rule

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

∫f′(x)g(f(x)) dx=G(f(x))+C\int f'(x)g(f(x))\,dx=G(f(x))+C

Recognize powers, exponentials and trigonometric functions with their derivative factors. Before using a longer technique, look for a direct rule and adjust for the inner derivative.

Common mistake

Dividing by the inner derivative is immediate only when that factor is constant or present in the integrand.

Step-by-step example

  1. Recognize the composite exponential.

    ∫e3x dx\int e^{3x}\,dx

  2. The inner derivative is 3.

    ddxe3x=3e3x\frac d{dx}e^{3x}=3e^{3x}

  3. Compensate for the factor.

    I=13e3x+CI=\frac13e^{3x}+C

Keep learning

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