Class CBSE Class 12 Mathematics Integrals Q #1291
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
Find: $\int x\sqrt{1+2x}dx$
Prev Next

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Substitution

Let $u = 1 + 2x$. Then, $du = 2dx$, so $dx = \frac{1}{2}du$. Also, $x = \frac{u-1}{2}$.

Step 2: Rewrite the integral

Substituting these into the integral, we get: $\int x\sqrt{1+2x}dx = \int \frac{u-1}{2}\sqrt{u}\frac{1}{2}du = \frac{1}{4}\int (u-1)\sqrt{u}du = \frac{1}{4}\int (u^{3/2} - u^{1/2})du$

Step 3: Integrate

Now, we integrate term by term: $\frac{1}{4}\int (u^{3/2} - u^{1/2})du = \frac{1}{4} \left( \int u^{3/2}du - \int u^{1/2}du \right) = \frac{1}{4} \left( \frac{u^{5/2}}{5/2} - \frac{u^{3/2}}{3/2} \right) + C = \frac{1}{4} \left( \frac{2}{5}u^{5/2} - \frac{2}{3}u^{3/2} \right) + C = \frac{1}{10}u^{5/2} - \frac{1}{6}u^{3/2} + C$

Step 4: Substitute back

Substitute $u = 1 + 2x$ back into the expression: $\frac{1}{10}(1+2x)^{5/2} - \frac{1}{6}(1+2x)^{3/2} + C$

Step 5: Simplify (Optional)

We can simplify this expression further by factoring out $(1+2x)^{3/2}$: $(1+2x)^{3/2} \left( \frac{1}{10}(1+2x) - \frac{1}{6} \right) + C = (1+2x)^{3/2} \left( \frac{3(1+2x) - 5}{30} \right) + C = (1+2x)^{3/2} \left( \frac{3+6x-5}{30} \right) + C = (1+2x)^{3/2} \left( \frac{6x-2}{30} \right) + C = \frac{1}{15}(3x-1)(1+2x)^{3/2} + C$

Final Answer: $\frac{1}{10}(1+2x)^{5/2} - \frac{1}{6}(1+2x)^{3/2} + C$ or $\frac{1}{15}(3x-1)(1+2x)^{3/2} + C$

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 concept of integration by substitution and apply it to solve the problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (integration by substitution) to find the integral.<\/span>
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of integration techniques, specifically integration by substitution, which is a standard topic in the syllabus.

More from this Chapter

SA
Evaluate: $\int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$
MCQ_SINGLE
\(\int\frac{1}{x(\log~x)^{2}}dx\) is equal to:
SA
Find: $\int\frac{2x+1}{(x+1)^{2}(x-1)}dx$
SA
Find: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx$
SA
26. Find: $\int\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}dx$
View All Questions