Chapter 7

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int x(\sin x+\cos x) d x $$

Evaluate each integral. $$ \int \frac{1}{x^{2}+9} d x $$

Use the fact that $$ \cot x=\frac{\cos x}{\sin x} $$ to evaluate $$ \int \cot x d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int \frac{\cos 2 x}{1+\sin 2 x} d x $$

Evaluate each integral. $$ \int \frac{1}{x^{2}+2 x+10} d x $$

The integral \(\int \ln x d x\) can be evaluated in two ways. (a) Write \(\ln x=1 \cdot \ln x\) and use integration by parts to evaluate the integral. (b) Use the substitution \(u=\ln x\) and integration by parts to evaluate the integral.

Evaluate each integral. $$ \int \frac{x}{x^{2}+4 x+5} d x $$

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

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

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

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. $$ \int_{1}^{2} x^{2} \ln x d x $$

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

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

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

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

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.

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 $$ t=\int \frac{d N}{a N \ln (b / N)} $$ Evaluate the integral (7.13), keeping \(a\) and \(b\) as unknown constants

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

Fish Growth von Bertalanffy's equation is used to model the growth of fish. The length of the fish, \(L(t)\), grows at a rate that depends on its current length \(L(t)\) (that is, big and small fish grow at different \mathrm{\\{} r a t e s ) . ~ The growth of a fish is modeled using the differential equation: $$ \frac{d L}{d t}=k\left(L_{\infty}-L\right) $$ where \(k\) and \(L_{\infty}\) are both positive constants. To solve the equation, we will learn in Chapter 8 that it is necessary to calculate the following integral: $$ t=\int \frac{d L}{k\left(L_{\infty}-L\right)} $$ Assume \(k=L_{\infty}=1 ;\) then evaluate the integral. $$ t=\int \frac{d L}{1-L} $$ Your answer will contain an unknown constant of integration.

Evaluate each integral. $$ \int \frac{1}{x^{2}-9} d x $$

Fish Growth von Bertalanffy's equation is used to model the growth of fish. The length of the fish, \(L(t)\), grows at a rate that depends on its current length \(L(t)\) (that is, big and small fish grow at different \mathrm{\\{} r a t e s ) . ~ The growth of a fish is modeled using the differential equation: $$ \frac{d L}{d t}=k\left(L_{\infty}-L\right) $$ where \(k\) and \(L_{\infty}\) are both positive constants. To solve the equation, we will learn in Chapter 8 that it is necessary to calculate the following integral: $$ t=\int \frac{d L}{k\left(L_{\infty}-L\right)} $$ Evaluate the integral (7.14), keeping \(k\) and \(L_{\infty}\) as unknown constants.

Evaluate each integral. $$ \int \frac{1}{x^{2}+16} d x $$

Evaluate each integral. $$ \int \frac{1}{x^{2}-x-2} d x $$

Evaluate each integral. $$ \int \frac{1}{x^{2}-x+2} d x $$

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

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

Evaluate each integral. $$ \int \frac{x^{2}+4}{x^{2}-4} d x $$

Evaluate each integral. $$ \int \frac{x^{4}+3}{x^{2}-4 x+3} d x $$

Evaluate each definite integral. $$ \int_{3}^{5} \frac{x-1}{(x+1)(x+2)} d x $$

Evaluate each definite integral. $$ \int_{3}^{5} \frac{x}{x-1} d x $$

Evaluate each definite integral. $$ \int_{0}^{1} \frac{x}{x^{2}+1} d x $$

Evaluate each definite integral. $$ \int_{1}^{2} \frac{x+1}{x^{2}+1} d x $$

Evaluate each definite integral. $$ \int_{2}^{3} \frac{1}{1-x} d x $$

Evaluate each definite integral. $$ \int_{2}^{3} \frac{1}{1-x^{2}} d x $$

Evaluate the definite integral. Hint: First integrate by parts to turn the integrand into a rational function. $$ \int_{0}^{1} \tan ^{-1} x d x $$

Evaluate the definite integral. Hint: First integrate by parts to turn the integrand into a rational function. $$ \int_{0}^{1} x \tan ^{-1} x d x $$