Question

evaluate each definite integral. show your work for f(x) and show your answer.

1. (int_{-1}^{1}(x^{4}-x^{3}-3x^{2}+1)dx)
2. (int_{2}^{7}(x^{2}-8x + 13)dx)
3. (int_{-2}^{-1}\frac{5}{x^{3}}dx)
4. (int_{-3}^{1}(x^{3}+2x^{2}-4x-6)dx)

Explanation:

Problem 6

Step1: Find antiderivative

$F(x) = \frac{x^5}{5} - \frac{x^4}{4} - x^3 + x$

Step2: Evaluate at bounds

$F(1) = \frac{1}{5} - \frac{1}{4} - 1 + 1 = -\frac{1}{20}$
$F(-1) = -\frac{1}{5} - \frac{1}{4} + 1 - 1 = -\frac{9}{20}$
$F(1)-F(-1) = -\frac{1}{20} - (-\frac{9}{20}) = \frac{8}{20} = \frac{2}{5}$

Step1: Find antiderivative

$F(x) = \frac{x^3}{3} - 4x^2 + 13x$

Step2: Evaluate at bounds

$F(7) = \frac{343}{3} - 196 + 91 = \frac{343}{3} - 105 = \frac{343 - 315}{3} = \frac{28}{3}$
$F(2) = \frac{8}{3} - 16 + 26 = \frac{8}{3} + 10 = \frac{38}{3}$
$F(7)-F(2) = \frac{28}{3} - \frac{38}{3} = -\frac{10}{3}$

Step1: Rewrite integrand

$5x^{-3}$

Step2: Find antiderivative

$F(x) = 5 \cdot \frac{x^{-2}}{-2} = -\frac{5}{2x^2}$

Step3: Evaluate at bounds

$F(-1) = -\frac{5}{2(1)} = -\frac{5}{2}$
$F(-2) = -\frac{5}{2(4)} = -\frac{5}{8}$
$F(-1)-F(-2) = -\frac{5}{2} - (-\frac{5}{8}) = -\frac{20}{8} + \frac{5}{8} = -\frac{15}{8}$

Answer:

$\frac{2}{5}$

Problem 7