Two Tricky Integration By Parts Examples

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0:00Rewrite x⁷√($1+x$⁴) as x⁴ times x³√($1+x$⁴). Choosing the first factor for differentiation makes the second factor integrable with a substitution.

3:00The substitution $1+x$⁴ has derivative 4x³, so it finds the antiderivative of x³√($1+x$⁴). Insert that result into integration by parts rather than treating the two methods as mutually exclusive.

6:00For eˣcos x, two applications of integration by parts return the original integral with a minus sign. Moving it across gives the antiderivative eˣ($\sin x+\cos x$)/2+C.

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