Math 3112
Test 1
Fall 1996

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The test has 7 problems on 4 pages.
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Problem 1 (15 points)   Find the following anti-derivatives:

1.

$$\int \sin 3x\, dx=$$

2.

$$\int \cot x\,dx=$$

3.

$$\int \sec 2x\, dx=$$

Problem 2 (15 points)   Compute the exact value of $$\int_2^5 \frac{t^2+2}{t^2}\,dt$$.

Problem 3 (15 points)   Compute $$\int t e^{-t}\, dt$$.

Problem 4 (15 points)   Compute $$\int \cos^4 x\,dx$$.

Problem 5 (15 points)  

1.

Explain why the area of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ (see the picture below) is given by the integral     $$2b\cdot \int_{-a}^{a}\sqrt{1-\frac{x^2}{a^2}}\,dx$$

2.

Compute the area of the ellipse!

Problem 6 (10 points)  

1.

Give numerical approximations to $$I=\int_2^8 \frac{-1}{\ln t}\, dt$$ using the Midpoint Rule with 50 steps MID(50), and the Trapezoidal Rule with 50 steps TRAP(50).

2.

Order the three quantities I, MID(50) and TRAP(50) according to their size.

Problem 7 (15 points)   Compute $$\int \frac{x^2-2x-1}{(1-x)(x-2)^2}\,dx$$.