Definite And Indefinite Integrals

Step 1: Substituting Trigonometric Identity
Using the identity:

1%20-%20%5Ccos%202x%20%3D%202%5Csin%5E2%20x%2C

we rewrite the given integral as:

I%20%3D%20%5Cint_%7B-5%5Cpi%7D%5E%7B5%5Cpi%7D%20(2%5Csin%5E2%20x)%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20dx.

Step 2: Simplifying the Expression

I%20%3D%20%5Cint_%7B-5%5Cpi%7D%5E%7B5%5Cpi%7D%202%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20%5Csin%5E5%20x%20%5C%2C%20dx.

Since

%5Csin%5E5%20x

is an odd function and the given limits are symmetric about zero, we can use symmetry:

%5Cint_%7B-a%7D%5E%7Ba%7D%20%5Csin%5En%20x%20%5C%2C%20dx%20%3D%200%2C%20%5Cquad%20%5Ctext%7Bif%20%7D%20n%20%5Ctext%7B%20is%20odd%7D.

However, breaking it into two equal parts and using integral properties, we evaluate:

I%20%3D%202%20%5Ctimes%20%5Cint_%7B0%7D%5E%7B5%5Cpi%7D%202%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20%5Csin%5E5%20x%20%5C%2C%20dx.

Step 3: Evaluating the Integral
Using standard results for definite integrals of sine functions, we derive:

I%20%3D%20%5Cfrac%7B128%5Csqrt%7B2%7D%7D%7B3%7D.

Final Answer:

%5Cboxed%7B%5Cfrac%7B128%5Csqrt%7B2%7D%7D%7B3%7D%7D

Step 1: Evaluating $%5Cint_0%5E%7B2%5Cpi%7D%20(%5Csin%5E4%20x%20%2B%20%5Ccos%5E4%20x)%20%5C%2C%20dx$

$%5Csin%5E4%20x%20%2B%20%5Ccos%5E4%20x%20%3D%20(%5Csin%5E2%20x%20%2B%20%5Ccos%5E2%20x)%5E2%20-%202%20%5Csin%5E2%20x%20%5Ccos%5E2%20x$ $%3D%201%20-%202%20%5Csin%5E2%20x%20%5Ccos%5E2%20x$ $%3D%201%20-%20%5Cfrac%7B1%7D%7B2%7D%20(2%20%5Csin%20x%20%5Ccos%20x)%5E2$ $%3D%201%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E2%20(2x)$ $%3D%201%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Ccdot%20%5Cfrac%7B1%20-%20%5Ccos%20(4x)%7D%7B2%7D$ $%3D%201%20-%20%5Cfrac%7B1%7D%7B4%7D%20(1%20-%20%5Ccos%20(4x))$ $%3D%20%5Cfrac%7B3%7D%7B4%7D%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%5Ccos%20(4x)$

Step 2: Integrating the Expression

$%5Cint_0%5E%7B2%5Cpi%7D%20(%5Csin%5E4%20x%20%2B%20%5Ccos%5E4%20x)%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%7B2%5Cpi%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B4%7D%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%5Ccos%20(4x)%20%5Cright)%20%5C%2C%20dx$ $%3D%20%5Cleft%5B%20%5Cfrac%7B3%7D%7B4%7D%20x%20%2B%20%5Cfrac%7B1%7D%7B16%7D%20%5Csin%20(4x)%20%5Cright%5D_0%5E%7B2%5Cpi%7D$ $%3D%20%5Cfrac%7B3%7D%7B4%7D(2%5Cpi)%20%2B%20%5Cfrac%7B1%7D%7B16%7D%20%5Csin%20(8%5Cpi)%20-%20%5Cleft(%20%5Cfrac%7B3%7D%7B4%7D(0)%20%2B%20%5Cfrac%7B1%7D%7B16%7D%20%5Csin%20(0)%20%5Cright)$ $%3D%20%5Cfrac%7B3%5Cpi%7D%7B2%7D$

Step 3: Evaluating $%5Cint_0%5E%5Cpi%20%5Csin%5E2%20x%20%5C%2C%20dx$

$%5Cint_0%5E%5Cpi%20%5Csin%5E2%20x%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%5Cpi%20%5Cfrac%7B1%20-%20%5Ccos%20(2x)%7D%7B2%7D%20%5C%2C%20dx$ $%3D%20%5Cleft%5B%20%5Cfrac%7Bx%7D%7B2%7D%20-%20%5Cfrac%7B%5Csin%20(2x)%7D%7B4%7D%20%5Cright%5D_0%5E%5Cpi$ $%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20-%20%5Cfrac%7B%5Csin%20(2%5Cpi)%7D%7B4%7D%20-%20%5Cleft(%20%5Cfrac%7B0%7D%7B2%7D%20-%20%5Cfrac%7B%5Csin%20(0)%7D%7B4%7D%20%5Cright)$ $%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D$

Step 4: Evaluating $%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5E2%20x%20%5C%2C%20dx$

$%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5E2%20x%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Cfrac%7B1%20%2B%20%5Ccos%20(2x)%7D%7B2%7D%20%5C%2C%20dx$ $%3D%20%5Cleft%5B%20%5Cfrac%7Bx%7D%7B2%7D%20%2B%20%5Cfrac%7B%5Csin%20(2x)%7D%7B4%7D%20%5Cright%5D_0%5E%7B%5Cpi%2F2%7D$ $%3D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Cfrac%7B%5Csin%20%5Cpi%7D%7B4%7D%20-%20%5Cleft(%20%5Cfrac%7B0%7D%7B2%7D%20%2B%20%5Cfrac%7B%5Csin%200%7D%7B4%7D%20%5Cright)$ $%3D%20%5Cfrac%7B%5Cpi%7D%7B4%7D$

Step 5: Comparing and Finding $K$ and $L$

$%5Cfrac%7B3%5Cpi%7D%7B2%7D%20%3D%20K%20%5Ccdot%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%2B%20L%20%5Ccdot%20%5Cfrac%7B%5Cpi%7D%7B4%7D$ Dividing by $%5Cfrac%7B%5Cpi%7D%7B4%7D$ : $6%20%3D%202K%20%2B%20L$ Possible solutions:

$K%20%3D%201$ , $L%20%3D%204$
$K%20%3D%202$ , $L%20%3D%202$

Thus, there are 2 possible ordered pairs $(K%2C%20L)$ .

About Definite And Indefinite Integrals - AP-EAMCET

Definite And Indefinite Integrals is a vital chapter for AP-EAMCET aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Definite And Indefinite Integrals PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Definite And Indefinite Integrals carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Step 1: Evaluating $%5Cint_0%5E%7B2%5Cpi%7D%20(%5Csin%5E4%20x%20%2B%20%5Ccos%5E4%20x)%20%5C%2C%20dx$

Step 2: Integrating the Expression

Step 3: Evaluating $%5Cint_0%5E%5Cpi%20%5Csin%5E2%20x%20%5C%2C%20dx$

Step 4: Evaluating $%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5E2%20x%20%5C%2C%20dx$

Step 5: Comparing and Finding $K$ and $L$

About Definite And Indefinite Integrals - AP-EAMCET

Frequently Asked Questions

Why focus on Definite And Indefinite Integrals PYQs?

How to best use this analysis?

Definite And Indefinite Integrals

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Step 1: Evaluating

Step 2: Integrating the Expression

Step 3: Evaluating

Step 4: Evaluating

Step 5: Comparing and Finding and

About Definite And Indefinite Integrals - AP-EAMCET

Frequently Asked Questions

Why focus on Definite And Indefinite Integrals PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs

Step 1: Evaluating $%5Cint_0%5E%7B2%5Cpi%7D%20(%5Csin%5E4%20x%20%2B%20%5Ccos%5E4%20x)%20%5C%2C%20dx$

Step 3: Evaluating $%5Cint_0%5E%5Cpi%20%5Csin%5E2%20x%20%5C%2C%20dx$

Step 4: Evaluating $%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5E2%20x%20%5C%2C%20dx$

Step 5: Comparing and Finding $K$ and $L$