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Chapter exploration

Techniques of integration

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Compare xex2xe^{x^2} with xe2xxe^{2x}. Before integrating, differentiate each exponent and decide whether its derivative accounts for the remaining factor. Try substitution u=x2u=x^2 in the first and parts with u=xu=x in the second. Explain what each choice simplifies, then differentiate both proposed answers. A familiar-looking product alone does not determine the method.

Related interactive example

Explore this related example, then return to the prediction task above.

Does (ln x)⁶/12 track the area?

Move the two bounds to select an interval. Compare its signed integral with the change in the antiderivative.

Integration boundsa = 1.00, x = 2.50

Move either handle to choose the interval. The graphs compare the signed integral with F(x) − F(a).

Try an example

How the shaded areas add up

The limits run left to right. Area above the axis adds; area below the axis subtracts.

Above-axis contribution
0.049
Below-axis contribution
0
Signed integral
0.049

All three bars share one scale. Positive contributions extend right of zero; negative contributions extend left.

Signed integral from 1.00 to x
0.049
F(x) − F(a)
0.049
agrees at displayed precision
Formula − integral
0
0.00% of the integral’s magnitude
Compare the running integral on a graph
  • Integrand
  • Numerical signed integral from the start to x
  • Part of the integrand above the axis
  • Part of the integrand below the axis
  • F(x) − F(start), from the antiderivative
Key idea

Differentiating (ln⁡x)612\frac{(\ln x)^6}{12} gives (ln⁡x)52x\frac{(\ln x)^5}{2x}: the dashed curve and the solid curve agree.

Ideas to try
  1. Move both bounds below 1: the integrand and signed integral are negative.
  2. Set the lower bound to 1 and the upper bound to ee: what is the integral?
Open the full interactive problem →
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