91Ó°ÊÓ

Evaluate. $$ \int_{0}^{\pi / 6} \frac{5}{2} \sin 2 x d x $$

Determine for which values of \(r\) the integral \(\int_{1}^{\infty} x^{r} d x\) is convergent and for which values of \(r\) it is divergent.

Graph the function and estimate the integral using a grapher. $$ \int_{0}^{\pi} \sin x d x $$

Estimate using a) the Trapezoid rule. b) Simpson's rule. $$ \int_{0}^{1} e^{-x^{2}} d x, \text { use } n=4 $$

Find the area of the region bounded by the given graphs. $$ y=\sqrt{1-x^{2}}, y=1-x^{2}, x=-1, x=1 $$

Find \(f\) such that: $$ f^{\prime}(x)=3 \sin 8 x, \quad f(0)=3 $$

a) Integrate \(\int x \sqrt[3]{x+2} d x\) using a substitution. b) Use integration by parts to compute \(\int x \sqrt[3]{x+2} d x\) c) Use algebra to show that your answers in parts (a) and (b) are equal.

Find \(f\) such that: $$ f^{\prime}(x)=5 e^{2 x}, f(0)=-10 $$

Consider the following [unctions: $$ \begin{array}{l} f(x)=3.8 x^{5}-18.6 x^{3} \\ g(x)=19 x^{4}-55.8 x^{2} \end{array} $$ a) Graph these functions in the window \([-4,4,-70,70]\), with Yscl \(=10\) b) Estimate the first coordinates \(a, b\), and \(c\) of the three points of intersection of the two graphs. c) Find the area between the curves on the interval \([a, b]\) d) Find the area between the curves on the interval \([b, c]\)

Evaluate using a substitution. (Be sure to check by differentiating!) $$ \int \frac{e^{\sqrt{w}}}{\sqrt{w}} d w $$