MATH 181  MIDTERM 1  FALL 1992

Professor Libgober NAME: (PRINT) SIGNATURE: SSN: T.A.: Find the following integrals:

1. ∫ x^{2}e^{−2x}dx

2. ∫ x sin^{2} x dx

MATH 181  MIDTERM 1  FALL 1992

3. ∫_{0}^{π/4} cos^{5} 2θ dθ.

4. Write \frac{(x + 1)(x + 2)}{(x − 1)^{3}(x^{2} − 4)(x^{2} + 4x + 5)^{2}} as sum of partial fractions with undetermined coefficients. Do not calculate the coefficients.

MATH 181  MIDTERM 1  FALL 1992

Calculate Integrals:

5. ∫ \frac{dx}{(x^{2} + 9)^{2}}

6. ∫ x^{2}\sqrt{1 − x^{2}} dx

MATH 181  MIDTERM 1  FALL 1992

7. ∫ sin^{3} x cos^{8} x dx 8. ∫_{0}^{3} \frac{dx}{x^{2} − x − 2}

MATH 181  MIDTERM 1  FALL 1992

9. ∫ \frac{dx}{x^{3} − 2}

10. Write down the approximation for ∫_{0}^{1} cos(x^{2})dx using trapezoidal rule with 5 intervals. Estimate the error. According to trapezoidal rule

∫_{a}^{b} f(x)dx ≅ \frac{Δx}{2} [f(x_{1}) + 2f(x_{2}) + … + 2f(x_{n}) + f(x_{n+1})]

and |E_{n}| ≤ \frac{M(b − a)^{3}}{12n^{2}} when M = max |f′′(x)| and a ≤ x ≤ b.