Integrate (i) [tex]$\int 3^x dx[/tex] (ii) [tex]$\int \ln x^2 dx[/tex].

Integrate 3 x using the formula ∫ a x d x = l n a a x ​ + C , which gives l n 3 3 x ​ + C .
Simplify ln x 2 to 2 ln ∣ x ∣ .
Integrate 2 ln ∣ x ∣ by parts, letting u = 2 ln ∣ x ∣ and d v = d x .
Obtain the final result: 2 x ln ∣ x ∣ − 2 x + C ​ .

Explanation

Problem Analysis We are asked to find the indefinite integrals of two functions: (i) 3 x and (ii) ln x 2 . We will use the appropriate integration techniques for each.

Integrating 3 x (i) To integrate 3 x , we use the formula ∫ a x d x = l n a a x ​ + C , where a is a constant. In this case, a = 3 . Therefore, the integral is:

Result of Integration of 3 x ∫ 3 x d x = ln 3 3 x ​ + C

Integrating ln x 2 (ii) To integrate ln x 2 , we first simplify the expression using the property of logarithms: ln x 2 = 2 ln ∣ x ∣ . Now we need to integrate 2 ln ∣ x ∣ . We will use integration by parts.

Applying Integration by Parts Let u = 2 ln ∣ x ∣ and d v = d x . Then, d u = x 2 ​ d x and v = x . Using the integration by parts formula, ∫ u d v = uv − ∫ v d u , we have:

Applying the Formula ∫ 2 ln ∣ x ∣ d x = 2 x ln ∣ x ∣ − ∫ x ⋅ x 2 ​ d x = 2 x ln ∣ x ∣ − ∫ 2 d x

Result of Integration of ln x 2 ∫ 2 ln ∣ x ∣ d x = 2 x ln ∣ x ∣ − 2 x + C

Final Result Therefore, the integral of ln x 2 is 2 x ln ∣ x ∣ − 2 x + C .


Examples
Imagine you're calculating the total growth of a bacterial population over time, where the growth rate is exponential, similar to 3 x . Integrating this growth rate gives you the total population size. Similarly, ∫ ln x 2 d x can model situations where the rate of change is logarithmic, such as in certain chemical reactions or economic models. Understanding these integrals helps predict cumulative effects in various real-world scenarios.

Answered by GinnyAnswer | 2025-07-08

To integrate 3 x , we apply the formula for exponential integrals, yielding l n 3 3 x ​ + C . For ln x 2 , after simplification to 2 ln ∣ x ∣ , we use integration by parts to get 2 x ln ∣ x ∣ − 2 x + C . Both results are important for understanding integration in calculus.
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Answered by Anonymous | 2025-07-15