Type 2 Problems Part 3 Problem No 3 - Definite Integration - Diploma Maths II

TL;DR

The video solves an integral problem using the property of complementary angles, resulting in the answer being PI/4.

Transcript

click the Bell icon to get latest videos from Ekeeda Hello friends in this video we are going to continue one more problem which is based for our second property so let us start with problem number 2 integral 0 to Pie by 2 under root sine X upon under root sin X plus under root cos x DX let us consider this given integral as I first and take it as ... Read More

Key Insights

• 🔺 The given integral is rewritten using the property of complementary angles.
• 💁 The integral simplifies to a form with sine and cosine terms.
• 🪜 The simplified integral is added to the original integral, canceling out common denominators.
• ❓ The final answer is found to be PI/4.
• 👨‍💼 Complementary angle formulas for sine and cosine are utilized.
• ❓ The formula for integral properties is mentioned.
• 🪜 The concept of adding integrals with the same denominator is explained.

Questions & Answers

Q: What is the given integral in the video?

The given integral in the video is ∫(0 to PI/2) √(sin x) / (√(sin x) + √(cos x)) dx.

Q: How is the integral simplified using the property of complementary angles?

By replacing x with PI/2 - x, the integral is written as ∫(0 to PI/2) √(sin(PI/2 - x)) / (√(sin(PI/2 - x)) + √(cos(PI/2 - x))) dx.

Q: What is the formula mentioned in the video regarding integral properties?

The formula mentioned is ∫(0 to a) f(x) dx = ∫(0 to a) f(a - x) dx.

Q: How is the simplified integral solved to find the final answer?

By adding the original integral with the simplified integral, the common denominators cancel, resulting in ∫(0 to PI/2) √(sin x) / (√(sin x) + √(cos x)) dx + ∫(0 to PI/2) √(cos x) / (√(cos x) + √(sin x)) dx = 2∫(0 to PI/2) dx = PI/2. Transferring variables, we get the final answer as PI/4.

Summary & Key Takeaways

• The video presents a problem involving an integral from 0 to PI/2 with a complicated numerator and denominator.

• By using the property of complementary angles, the integral is rewritten in terms of sine and cosine, simplifying the expression.

• The simplified integral is then solved, resulting in the answer PI/4.