91Ó°ÊÓ

In Problems 49-54, use integration by parts to derive the given formula. \(\begin{aligned} \int \cos 5 x \sin 7 x d x &=\\\ &-\frac{7}{24} \cos 5 x \cos 7 x-\frac{5}{24} \sin 5 x \sin 7 x+C \end{aligned}\)

In Problems 1-54, perform the indicated integrations. \(\int \frac{d x}{\sqrt{16+6 x-x^{2}}}\)

The Law of Mass Action in chemistry results in the differential equation $$ \frac{d x}{d t}=k(a-x)(b-x), \quad k>0, \quad a>0, \quad b>0 $$ where \(x\) is the amount of a substance at time \(t\) resulting from the reaction of two others. Assume that \(x=0\) when \(t=0\). (a) Solve this differential equation in the case \(b>a\). (b) Show that \(x \rightarrow a\) as \(t \rightarrow \infty\) (if \(b>a\) ). (c) Suppose that \(a=2\) and \(b=4\), and that 1 gram of the substance is formed in 20 minutes. How much will be present in 1 hour? (d) Solve the differential equation if \(a=b\).

In Problems 1-54, perform the indicated integrations. \(\int \frac{x+1}{9 x^{2}+18 x+10} d x\)

In Problems 49-54, use integration by parts to derive the given formula. \(\int e^{\alpha z} \sin \beta z d z=\frac{e^{\alpha z}(\alpha \sin \beta z-\beta \cos \beta z)}{\alpha^{2}+\beta^{2}}+C\)

The differential equation $$ \frac{d y}{d t}=k(y-m)(M-y), y(0)=y_{0} $$ with \(k>0\) and \(0 \leq m

In Problems 49-54, use integration by parts to derive the given formula. \(\int e^{\alpha z} \cos \beta z d z=\frac{e^{\alpha z}(\alpha \cos \beta z+\beta \sin \beta z)}{\alpha^{2}+\beta^{2}}+C\)

In Problems 1-54, perform the indicated integrations. \(\int \frac{3-x}{\sqrt{16+6 x-x^{2}}} d x\)

In Problems 49-54, use integration by parts to derive the given formula. \(\int x^{\alpha} \ln x d x=\frac{x^{\alpha+1}}{\alpha+1} \ln x-\frac{x^{\alpha+1}}{(\alpha+1)^{2}}+C, \alpha \neq-1\)

As a model for the production of trypsin from trypsinogen in digestion, biochemists have proposed the model $$ \frac{d y}{d t}=k(A-y)(B+y) $$ where \(k>0, A\) is the initial amount of trypsinogen, and \(B\) is the original amount of trypsin. Solve this differential equation.