91Ó°ÊÓ

Evaluate each integral. $$ \int \frac{1}{(x+1)\left(x^{2}+4\right)} d x $$

In Problems 63-68, evaluate each definite integral. $$ \int_{-1}^{0} \frac{2}{1+x^{2}} d x $$

Evaluate each integral. $$ \int \frac{2 x^{2}+x+5}{x\left(x^{2}+2 x+5\right)} d x $$

In Problems 63-68, evaluate each definite integral. $$ \int_{1}^{2} x^{2} \ln x d x $$

Evaluate each integral. $$ \int \frac{1}{x^{2}\left(x^{2}+1\right)} d x $$

In Problems 63-68, evaluate each definite integral. $$ \int_{0}^{\pi / 2} e^{x} \sin x d x $$

Evaluate each integral. $$ \int \frac{x^{2}+2 x}{(x+1)\left(x^{2}+2 x+2\right)} d x $$

In Problems 63-68, evaluate each definite integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(1+\tan ^{2} x\right) d x $$

Evaluate each integral. $$ \int \frac{1}{(x=3)(x+2)} d x $$

The Gompertz equation is used to model the growth of a tumor. We will study it in Chapter \(8 .\) In this model the number of cells \(N(t)\) in a tumorgrows over time at a rate that depends on \(N\), that is, tumors of different sizes grow at different rates, producing a differential equation: $$ \frac{d N}{d t}=a N \ln (b / N) $$ where a and b are positive constants that depend on the type of tumor, whether the tumor is being treated, and on the kind of treatment. In Chapter 8 we will see that the solution to this equation is given by evaluating the integral $$ t=\int \frac{d N}{a N \ln (b / N)} $$ Assume \(a=b=1\); then evaluate the integral \(t=\int \frac{d N}{N \ln (1 / N)}\). Your answer will contain an unknown constant of integration.