Substitution in integrals evaluate the definite integral

Question

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(e^5x/1+e^5x)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.