Solutions

Problem 1 (15 points)   Compute the exact value of

Solution:

Problem 2 (10 points)   Compute the exact value of

Solution: Use Integration by Parts: :

Problem 3 (15 points)   Find

Solution: We will use Integration by Parts twice. First let :

Now transform the integral on the right via :

The integrals on the left and the right side are identical. Consequently

i.e.

Problem 4 (10 points)   Find

Solution: First perform polynomial division:

The denominator is a repeated linear factor, so the setup for the partial fractions method is as follows:

Solving for the unknowns yields A=4, B=-7. Thus

Problem 5 (10 points)   Find

Solution: We will use the substitution , and the Pythagorean Theorem .

Problem 6 (15 points)   Find

Solution: We will first substitute y=x-1.

Next we use Integration by Parts: . Thus

Using the substitution w=1-y2, it is easy to see that . Resubstituting we obtain

Problem 7 (10 points)   Find

Solution: Since , we use the trigonometric substitution .

Problem 8 (15 points)   Find

Solution: We split the integral into 2 pieces:

For the first part we use the substitution u=(x-1)2+9, for the second part .

[Calculus] [CyberExam]

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Helmut Knaust