INTEGRAL CALCULUS

HOMEWORK 6

Problem 1:

Evaluate the integral

∫ x2 − x + 1

x − 2 dx using polynomial long division.

a.) Use long division to break apart the integrand into a polynomial and a remainder.

b.) Evaluate the broken up integrand you found above.

Problem 2:

Rewrite the integrand of the integral

∫ 1

x2 + 3x − 3 dx by completing the square.

a.) Complete the square for the polynomial in the denominator.

b.) (Extra) Use a trigonometric substitution to evaluate the integral (Hint: use two

successive u-subs).

1

2 HOMEWORK 6

Problem 3:

Determine whether the following statements are true or false and give an explanation or

counterexample:

a.)

∫ uv′dx =

(∫ udx

)(∫ v′dx

) .

b.)

∫ uv′dx = uv −

∫ vu′dx.

c.)

∫ vdu = uv −

∫ udv.

Problem 4:

Evaluate the integral

∫ π 0

x cos(x)dx:

HOMEWORK 6 3

Problem 5:

Evaluate the integral

∫ x ln2(x)dx:

Problem 6:

Use trigonometric identities to evaluate the integral

∫ sin3(x) cos2(x)dx:

Problem 7:

Use a trigonometric substitution to evaluate the integral

∫ x2

16 + x2 dx: