Welcome to Poly Learn Hub! In this article, we’ll solve question from the Integration chapter (Exercise 1.1, Q1 part i) for 2nd Semester Polytechnic Mathematics.
Question
Integrate:
∫ 1 / (x√x) dx
Step-by-Step Solution
Step 1: Simplify the Expression
We are given:
∫ 1 / (x√x) dx
We can write √x as x1/2, and multiply it with x in the denominator:
= ∫ 1 / (x * x1/2) dx
Now apply the rule of exponents:
x * x1/2 = x3/2
So the expression becomes:
= ∫ x-3/2 dx
Step 2: Apply the Power Rule
We use the formula:
∫ xⁿ dx = xn+1 / (n+1) + C, for n ≠ –1
Here, n = –3/2
= x-3/2 + 1 / (-3/2 + 1) + C
= x-1/2 / (-1/2) + C
Step 3: Simplify the Result
= -2x-1/2 + C
= -2 / √x + C
Final Answer
∫ 1 / (x√x) dx = -2 / √x + C
Concepts Used in This Problem
- Converting square roots into fractional exponents
- Simplifying expressions using exponent rules
- Applying the power rule for integration
- Understanding the constant of integration (C)
Learn More
This question is part of Exercise 1.1 in the Integration chapter of Mathematics – Semester 2 (Polytechnic Diploma syllabus).
Such questions are commonly asked in:
- Board exams
- Internal assessments
- Viva questions
- Competitive entrance exams (for lateral entry/diploma upgrades)
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Final Words
We hope this step-by-step explanation helped you understand how to approach integration problems like this. Keep practicing more problems from Exercise 1.1 to strengthen your fundamentals.
Stay connected with Poly Learn Hub for more such detailed solutions, notes, and resources for your polytechnic diploma journey!
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