Integration By Change Of Variables (Integration By Substitution)

E. INTEGRATION BY CHANGE OF VARIABLES (Integration by Substitution)

Suppose an integral has the form

When we make the substitution u = g(x), du = g'(x) dx, the integral becomes ∫ f(u) du. This new integral is often simpler than the original one.

Example 6

Example 7

Clue: If an integral invokes √x, try the substitution u = √x, dx = 2u du. If an integral involves , try

Example 8

Clue If an integral has the form ∫f(ax² + b)x dx, try u = ax² + b, du = ax dx.

If the derivatives in formulas I-XII are solidly memorized, then one can often recognize integrals of the form ∫ f(g(x))g'(x) dx and find the right substitution. Here are three more clues.

Clue Given ∫ f(a^x)a^x dx, put a^x = e^{x ln a} and try the substitution u = a^x,du = (ln a)a^x dx.

Clue Given ∫ f(sin x) cos x dx, try u = sin x, du = cos x dx.