Chapter 12

Evaluate the following definite integrals. $$ \int_{-1}^{0}\left(1+x-x^{3}\right) d x $$

Evaluate the following definite integrals. $$ \int_{6}^{11}(x-2)^{1 / 2} d x $$

Evaluate the following definite integrals. $$ \int_{1}^{3} \frac{t}{t+1} d t $$

Evaluate the following definite integrals. $$ \int_{0}^{6}|x-3| d x $$

Evaluate the following definite integrals. If \(\int_{0}^{k}(6 x-1) d x=4,\) find \(k\).

Evaluate the following definite integrals. $$ \int_{0}^{\pi} \frac{\sin x}{\sqrt{1+\cos x}} d x $$

Evaluate the following definite integrals. If \(f^{\prime}(x)=g(x)\) and \(g\) is a continuous function for all real values of \(x\) express \(\int_{1}^{2} g(4 x) d x\) in terms of \(f\).

Evaluate the following definite integrals. $$ \int_{\ln 2}^{\ln 3} 10 e^{x} d x $$

Evaluate the following definite integrals. $$ \int_{e}^{e^{2}} \frac{1}{t+3} d t $$

Evaluate the following definite integrals. If \(f(x)=\int_{-\pi / 4}^{x} \tan ^{2}(t) d t,\) find \(f^{\prime}\left(\frac{\pi}{6}\right)\).

Evaluate the following definite integrals. $$ \int_{-1}^{1} 4 x e^{x^{2}} d x $$

Evaluate the following definite integrals. $$ \int_{-\pi}^{\pi}\left(\cos x-x^{2}\right) d x $$

Evaluate the following definite integrals. Find \(k\) if \(\int_{0}^{2}\left(x^{3}+k\right) d x=10\).

Evaluate the following definite integrals. Evaluate \(\int_{-1.2}^{3.1} 2 \theta \cos \theta d \theta\) to the nearest 100 th

Evaluate the following definite integrals. If \(y=\int_{1}^{x^{3}} \sqrt{t^{2}+1} d t,\) find \(\frac{d y}{d x}\).

Use a midpoint Riemann sum with four subdivisions of equal length to find the approximate value of \(\int_{0}^{8}\left(x^{3}+1\right) d x\).

Given \(\int_{-2}^{2} g(x) d x=8\) and \(\int_{0}^{2} g(x) d x=3,\) find (a) \(\int_{-2}^{0} g(x) d x\) (b) \(\int_{2}^{-2} g(x) d x\) (c) \(\int_{0}^{-2} 5 g(x) d x\) (d) \(\int_{-2}^{2} 2 g(x) d x\)

Evaluate the following definite integrals. Evaluate \(\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}} .\)

Evaluate the following definite integrals. Find \(\frac{d y}{d x}\) if \(y=\int_{\cos x}^{\sin x}(2 t+1) d t\).

Evaluate the following definite integrals. Let \(f\) be a continuous function defined on [0,30] with selected values as shown below: $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline f(x) & 1.4 & 2.6 & 3.4 & 4.1 & 4.7 & 5.2 & 5.7 \\ \hline \end{array}$$ Use a midpoint Riemann sum with three subdivisions of equal length to find the approximate value of \(\int_{0}^{30} f(x) d x\).

(Calculator) indicates that calculators are permitted. Evaluate \(\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-4}}{3 x-9}\)

(Calculator) indicates that calculators are permitted. Find \(\frac{d y}{d x}\) at \(x=3\) if \(y=\ln \left|x^{2}-4\right|\).

(Calculator) Given the equation \(9 x^{2}+4 y^{2}-18 x+16 y=11,\) find the points on the graph where the equation has a vertical or horizontal tangent.