Q. 14 - Calculus

Question

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

Step-by-Step Solution

Answer

The three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x) is $\int \frac{8x}{x^2+1}dx,\int \frac{1}{3x+1}dx \mathrm{and} \int \frac{e^x}{2-e^x}dx$.

Step 1. Given Information

Find three integrals in Exercises 21–70 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)

$\int \frac{8x}{x^2+1}dx$

In his integration if we choose $x^2+1=u$ then $\frac{du}{dx}=2x$ , so $\frac{1}{2}du=xdx$. We can write the integral as $\int \frac{8x}{x^2+1}dx=\frac{8}{2}\int \frac{du}{u}=4\int \frac{du}{u}$ 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)

$\int \frac{1}{3x+1}dx$

In his integration if we choose $3x+1=u$ then $\frac{du}{dx}=3$, so $\frac{1}{3}du=dx$. We can write the integral as $\int \frac{1}{3x+1}dx=\frac{1}{3}\int \frac{1}{u}du$ 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)

$\int \frac{e^x}{2-e^x}dx$

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