Reference no: EM131082469
Substitution in Integrals - Math 1A, section 103
1. Find the integral ∫x2dx.
2. Compute ∫tan2(x) sec2(x) dx.
3. Evaluate the definite integral 0∫1(e5x/1+e5x)dx.
4. (a) Evaluate ∫sin3(x) dx, by writing sin2(x) = 1 - cos2(x) and making a substitution.
(b) Use a similar method to evaluate ∫sin5(x) dx.
(c) What goes wrong when you try to use this method on ∫sin2(x) dx?
(d) Recall the trig identities sin2(x) = 1-cos(2x)/2 and cos2(x) = 1+cos(2x)/2. Use these to compute ∫sin2(x) dx.
5. Pretend it is 5000 BC and no humans yet know the area of a circle. How could you figure out the area of a circle of radius 1 using integrals? Try to evaluate the integral using "reverse u-substitution", that is, express your integral using the variable u, and make the substitution u = sin(x) to express it as an integral over x.
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