Q. 14 Find three integrals in Exercise... [FREE SOLUTION]

Chapter 5: Q. 14 (page 417)

Find three integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x).

Short Answer

Expert verified

Thethree integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x) is $鈭� \frac{8 x}{x^{2} + 1} d x, 鈭� \frac{1}{3 x + 1} d x and 鈭� \frac{e^{x}}{2 - e^{x}} d x$ .

Step by step solution

Step 1. Given Information

Find three integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x).

Step 2. The first integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

$鈭� \frac{8 x}{x^{2} + 1} d x$

In his integration if we choose $x^{2} + 1 = u$ then $\frac{d u}{d x} = 2 x$ , so $\frac{1}{2} d u = x d x$ . We can write the integral as $鈭� \frac{8 x}{x^{2} + 1} d x = \frac{8}{2} 鈭� \frac{d u}{u} = 4 鈭� \frac{d u}{u}$ which we know how to integrate.

Step 3. The second integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

$鈭� \frac{1}{3 x + 1} d x$

In his integration if we choose $3 x + 1 = u$ then $\frac{d u}{d x} = 3$ , so role="math" $\frac{1}{3} d u = d x$ . We can write the integral as role="math" localid="1648788212772" $鈭� \frac{1}{3 x + 1} d x = \frac{1}{3} 鈭� \frac{1}{u} d u$ which we know how to integrate.

Step 4. The third integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

$鈭� \frac{e^{x}}{2 - e^{x}} d x$

In his integration if we choose $2 - e^{x} = u$ then $\frac{d u}{d x} = - e^{x}$ , so $- d u = e^{x} d x$ . We can write the integral as $鈭� \frac{e^{x}}{2 - e^{x}} d x = - 鈭� \frac{1}{u} d u$ which we know how to integrate.

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91影视

Find three integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x).

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The first integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

Step 3. The second integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

Step 4. The third integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x)

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