91Ó°ÊÓ

Evaluate the integrals. $$\int \frac{x+4}{x^{2}+4} d x$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\theta \sin \left(\log _{7} \theta\right)$$

Which one is correct, and which one is wrong? Give reasons for your answers. a. \(\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-3}=\lim _{x \rightarrow 3} \frac{1}{2 x}=\frac{1}{6}\) b. \(\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-3}=\frac{0}{6}=0\)

Derive the formula \(\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})\) for all real \(x\) Explain in your derivation why the plus sign is used with the square root instead of the minus sign.

Evaluate the integrals. $$\int \frac{t-2}{t^{2}-6 t+10} d t$$

Which one is correct, and which one is wrong? Give reasons for your answers. $$\begin{aligned} a. \lim _{x \rightarrow 0} \frac{x^{2}-2 x}{x^{2}-\sin x} &=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x} \\ &=\lim _{x \rightarrow 0} \frac{2}{2+\sin x}=\frac{2}{2+0}=1 \end{aligned}$$ $$\text { b. } \lim _{x \rightarrow 0} \frac{x^{2}-2 x}{x^{2}-\sin x}=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x}=\frac{-2}{0-1}=2$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)$$

Evaluate the integrals. $$\int \frac{x^{2}+2 x-1}{x^{2}+9} d x$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{10} e^{x}$$

Find the lengths of the following curves. a. \(y=\left(x^{2} / 8\right)-\ln x, \quad 4 \leq x \leq 8\) b. \(x=(y / 4)^{2}-2 \ln (y / 4), \quad 4 \leq y \leq 12\)