Q. 2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Five integrals for which u-substitution is a better strategy than integration by parts. List a good choice for u in each case.

(b) Five integrals for which integration by parts is a better strategy than u-substitution. List good choices for u and dv in each case.

(c) Three integrals that we cannot integrate with only the techniques we have learned so far.

Step-by-Step Solution

Answer

(a) The integrals for which the u-substitution is better strategy than integration by parts is:

$$\begin{gathered} \int \frac{\ln x}{x}dx where u=\ln x \\ \int \frac{\cos (\ln x)}{x}dx where u=\ln x \\ \int \frac{x}{x^2+1}dx where u=x^2+1 \\ \int \frac{1}{x \ln x}dx where u=\ln x \\ \int \frac{e^x}{csc e^x}dx where u=e \end{gathered}$$

(b) The integrals for which the integration by parts is better strategy than u-substitution is:

$$\begin{gathered} \int x{e}^x dx, where u=x, dv=e^{x}dx \\ \int x\ln x dx, where u=x, dv=\ln dx \\ \int \frac{x}{e^x} dx, where u=x, dv=\frac{1}{e^x} dx \\ \int x.\cos x dx, where u=x, dv=\cos x dx \\ \int \ln \left(x^3\right) dx, where u=\ln x, dv= dx \end{gathered}$$

(c) The integrals that cannot be integrate are:

$$\begin{gathered} \int x. cs{c}^{2}x dx \\ \int x^{5}csc x^3 dx \\ \int \frac{x^2}{\sqrt{x-1}} dx \end{gathered}$$

Part (a) Step 1. Given Information.

u-substitution is a better strategy than integration by parts

Part (a) Step 2. Examples for such methods.

The examples for which u-substitution is better strategy than integration by parts is:

$$\begin{gathered} \int \frac{\ln x}{x}dx where u=\ln x \\ \int \frac{\cos (\ln x)}{x}dx where u=\ln x \\ \int \frac{x}{x^2+1}dx where u=x^2+1 \\ \int \frac{1}{x \ln x}dx where u=\ln x \\ \int \frac{e^x}{csc e^x}dx where u=e^x \end{gathered}$$

Part (b) Step 1. Examples for such methods.

The examples for which integration by parts is better strategy than u-substitution is:

$$\begin{gathered} \int x{e}^x dx, where u=x, dv=e^{x}dx \\ \int x\ln x dx, where u=x, dv=\ln dx \\ \int \frac{x}{e^x} dx, where u=x, dv=\frac{1}{e^x} dx \\ \int x.\cos x dx, where u=x, dv=\cos x dx \\ \int \ln \left(x^3\right) dx, where u=\ln x, dv= dx \end{gathered}$$

Part (c) Step 1. Examples for such methods.

The examples that cannot be integrated with the help of these methods is:

$$\begin{gathered} \int x. cs{c}^{2}x dx \\ \int x^{5}csc x^3 dx \\ \int \frac{x^2}{\sqrt{x-1}} dx \end{gathered}$$