# Substitution

You can use substitution to convert a complicated integral into a simpler one. In these problems, I'll let u equal some convenient x-stuff --- say . To complete the substitution, I must also substitute for . To do this, compute , so . Then .

Example. Compute .

Example. Compute .

Example. Later on, I'll derive the integration formula

Use this formula to compute .

Example. Compute .

Notice that in the second step in the last example, the x's cancelled out, leaving only u's. If the x's had failed to cancel, I wouldn't have been able to complete the substitution.

But what made the x's cancel? It was the fact that I got an x from the derivative of . This leads to the following rule of thumb.

Substitute for something whose derivative is also there.

Example. Compute .

Example. Compute .

Example. Compute .

Example. Compute .

Example. Compute .

Example. Compute .

Example. Compute .

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