Solving the Integral \(\rm \int e^{\left(2\ln x+\ln x^2\right)}dx\)

We are asked to evaluate the integral \(\rm \int e^{\left(2\ln x+\ln x^2\right)}dx\). To solve this, we first need to simplify the expression in the exponent using properties of logarithms.

Simplifying the Exponent

The exponent is \(\rm 2\ln x + \ln x^2\). We can use the following logarithm properties:

• The power rule: \(\rm a \ln b = \ln b^a\)
• The product rule: \(\rm \ln A + \ln B = \ln (AB)\)

Applying the power rule to the first term:

\(\rm 2\ln x = \ln x^2\)

Now substitute this back into the exponent expression:

\(\rm 2\ln x + \ln x^2 = \ln x^2 + \ln x^2\)

Using the product rule for logarithms:

\(\rm \ln x^2 + \ln x^2 = \ln (x^2 \cdot x^2)\)

Simplifying the term inside the logarithm:

\(\rm x^2 \cdot x^2 = x^{2+2} = x^4\)

So, the simplified exponent is \(\rm \ln x^4\).

Simplifying the Integrand

Now, the original expression inside the integral becomes:

\(\rm e^{\left(2\ln x+\ln x^2\right)} = e^{\ln x^4}\)

Using the property that \(\rm e^{\ln A} = A\), we can simplify further:

\(\rm e^{\ln x^4} = x^4\)

Thus, the integral we need to evaluate is \(\rm \int x^4 dx\).

Evaluating the Integral

Now we evaluate the simplified integral \(\rm \int x^4 dx\). We use the power rule for integration:

\(\rm \int x^n dx = \frac{x^{n+1}}{n+1} + c\), where \(n \neq -1\) and \(c\) is the constant of integration.

In this case, \(n=4\). Applying the power rule:

\(\rm \int x^4 dx = \frac{x^{4+1}}{4+1} + c\)

\(\rm \int x^4 dx = \frac{x^5}{5} + c\)

Final Result of the Integration

The value of the integral \(\rm \int e^{\left(2\ln x+\ln x^2\right)}dx\) is \(\rm \frac{x^5}{5}+c\).

Revision Table: Integration and Logarithm Properties

Understanding the properties of logarithms and basic integration rules is crucial for solving such problems.

Additional Information on Integral Evaluation

When evaluating integrals, especially those involving exponential and logarithmic functions, simplifying the expression before integrating is often the key. Using the properties of logarithms and the inverse relationship between \(e^x\) and \(\ln x\) can greatly simplify the integrand.

The power rule for integration is one of the most fundamental rules. It applies to any term of the form \(x^n\), as long as \(n\) is not equal to \(-1\). If \(n = -1\), the integral \(\int x^{-1} dx = \int \frac{1}{x} dx = \ln |x| + c\).

Remember that the constant of integration, \(c\), is always added to the result of an indefinite integral because there are infinitely many functions that have the same derivative (they differ only by a constant).