Evaluate the following integral: \$ \int ("ln"("x"))/"x" "dx"\$.

Let’s consider the integral as follows:

\$ \int ("ln"("x"))/"x" "dx"\$

Using the technique of integration by parts, we choose

\$ "u" = "ln"("x") \$ and \$"dv" = (1/"x") "dx" \$

This gives us

\$ "du" = (1/"x")dx \$

\$ "v" = \int (1/"x") "dx" = "ln"| x | \$

Applying the integration by parts formula:

\$ \int "u" "dv" = "uv" - \int "v" "du" \$

We can simplify the integral on the right-hand side now

\$ \int ("ln"("x"))/"x" "dx" = "ln"("x") times "ln"| "x" | - \int "ln"| "x" | times (1/"x") dx \$

\$ \int ("ln"("x"))/"x" "dx" = "ln"("x") times "ln"| "x" | - \int ("ln"| "x" |) / "x" "dx" \$

The RHS(right-hand side) integral is the same as the original. Hence, we can write the equation as follows. To simplify, we add the integral to both sides.

\$ \int ("ln"("x"))/"x" "dx" = "ln"("x") times "ln"| "x" | - \int ("ln"("x")) / "x" "dx" \$

\$ \int ("ln"("x"))/"x" "dx" + \int ("ln"("x")) / "x" "dx" = "ln"("x") times "ln"| "x" | - \int ("ln"("x")) / "x" "dx" + \int ("ln"("x")) / "x" "dx"\$

After simplification, the equation is divided by 2 after adding the same value to both sides of the equation

\$ 2\int ("ln"("x"))/"x" "dx" = "ln"("x") times "ln"| "x" | \$

\$ \cancel2/\cancel2 \int ("ln"("x")/"x") "dx" = ("ln"("x") times "ln"| "x" | )/2 + "C" \$

The \$ ln| x | \$ is nothing but \$ ln(x) \$

\$ \int ("ln"("x"))/"x" "dx" = (("ln"("x"))^2 )/2 + "C" \$

So, the value of the integral \$ \int ("ln"("x") )/"x" "dx"\$ is \$ (("ln"("x"))^2 )/2 + C\$ , where C is the constant of integration.

This is incorrect because the term \$("ln"("x"))^2\$ is wrong, it should be \$(1/2)("ln"("x"))^2\$to rectify the error

\$(ln("x"))^2 + C \$

This answer is correct as it precisely represents the antiderivative for the given integral

\$(1/2)(ln("x"))^2 + C \$

This is incorrect because the term \$2(ln("x"))^2\$ is wrong, it should be \$(1/2)(ln("x"))^2\$to rectify the error

\$2(ln("x"))^2 + C \$

Option D, \$2ln("x")+C\$, is incorrect because the integral doesn’t result in \$2ln("x")\$

\$2ln("x") + C \$

Option

B

Can you summarize what you’ve understood in the above steps?