Q1319: Find: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}...

Class CBSE Class 12 Mathematics Integrals Q #1319

KNOWLEDGE BASED

UNDERSTAND

3 Marks 2024 AISSCE(Board Exam) SA

Find: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx$

Prev Next

AI Tutor Explanation

Detailed Solution

Step 1: Identify the form

The integral is of the form $\int e^x [f(x) + f'(x)] dx$, which equals $e^x f(x) + C$. We need to manipulate the given expression to fit this form.

Step 2: Rewrite the integral

Let's rewrite the integral as: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx = \int e^x \left[ \frac{1}{(1+x^2)^{3/2}} + \frac{x}{(1+x^2)^{1/2}} \right] dx$

Step 3: Identify f(x) and f'(x)

Let $f(x) = \frac{x}{\sqrt{1+x^2}}$. Then we need to find $f'(x)$. $f'(x) = \frac{d}{dx} \left( \frac{x}{\sqrt{1+x^2}} \right)$

Step 4: Calculate f'(x)

Using the quotient rule: $f'(x) = \frac{\sqrt{1+x^2} \cdot 1 - x \cdot \frac{1}{2}(1+x^2)^{-1/2} \cdot 2x}{1+x^2} = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{1+x^2} = \frac{1+x^2 - x^2}{(1+x^2)^{3/2}} = \frac{1}{(1+x^2)^{3/2}}$

Step 5: Apply the formula

Now we have $\int e^x [f(x) + f'(x)] dx = \int e^x \left[ \frac{x}{\sqrt{1+x^2}} + \frac{1}{(1+x^2)^{3/2}} \right] dx$. Since $f(x) = \frac{x}{\sqrt{1+x^2}}$ and $f'(x) = \frac{1}{(1+x^2)^{3/2}}$, the integral is $e^x f(x) + C = e^x \frac{x}{\sqrt{1+x^2}} + C$

Final Answer: $e^{x}\frac{x}{\sqrt{1+x^{2}}}+C$

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the form of the integral $\int e^x [f(x) + f'(x)] dx$ and manipulate the given expression to fit this form.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply the procedure of identifying the form of the integral, finding the derivative, and then applying the formula.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question is directly based on the standard integration techniques taught in the textbook.