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 $(K%2C%20L)$ .

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 $L$

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

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 $L$