Question

7. (2 points)Evaluate the following definite integrals. a. \( \int_{-1}^{3}\left(4 x^{3}-2 x+1\right) d x \) b. \( \int_{2}^{5} e^{x} d x \) c. \( \int_{1}^{3} \frac{1}{x} d x \)

Answer 1

The given integrals are: a. ∫-14x3−2xdx b. ∫2e5xdx c. ∫11/xdxa. ∫−14x3−2xdxWe have to apply the power rule to evaluate this integral.Let u=4x3−2x+1The derivative of u, du is equal to 12x2−2dx∫−14x3−2xdx=14∫du=14u+C14(4x3−2x+1)+C=a polynomial in x+b.∫2e5xdxWe have to apply the formula for the integral of ex from a to b, where a=2 and b=5.∫2e5xdx=e5−e2=a number.∫11/xdxWe have to apply the rule for the integral of a power function.∫11/xdx=ln|x|∣13=ln(3)−ln(1)=ln(3)Answers:a. ∫-14x3−2xdx=14(4x3−2x+1)+C=a polynomial in x+b.b. ∫2e5xdx=e5−e2=a number.c. ∫11/xdx=ln|x|∣13=ln(3)−ln(1)=ln(3).