Question

Please don't solve this

$\int \: sin {}^{2} x \: dx$
Explain properly no copy..​

Answer 1

$\star \: \: { \bold{ \underline{ \red{ \underline{Integration}}}}} \implies$

$\int \: sin {}^{2} x \: dx \:$

Here , we shall find the integral by using trigonometric identity .

${ \boxed{ \boxed{ \bold{\: Cos \: 2 \:A \: = \: 1 - 2 Sin {}^{2} \: A}}}}$

From here we have ,

$\implies \: 2 \: Sin {}^{2} A \: = 1 - Cos \: 2A \\ \\ \implies \: \: Sin {}^{2} A \: = \dfrac{1 -Cos \: 2A }{2}$

Hence , substitute this value of Sin² A in the given integral :

$\int \: sin {}^{2} x \: dx \: = \int \: \dfrac{1 - Cos \: 2x }{2}$

Separating the integral ,

$\implies \: \int \: \dfrac{1}{2} dx \: - \int \: \dfrac{Cos \: 2x}{2} dx \\ \\ \implies \: \dfrac{x}{2} - \dfrac{sin \: 2x }{4} + C \:$

Which is the required answer !

Answer 2

Explanation:

Answer:

sin²x cos²x dx = (1/8)x - (1/8) sinx cos³x + (1/8) sin³x cosx + C

Step-by-step explanation:

Recall the double-angle identity sin(2x) = 2 sinx cosx;  

Hence:  

sinx cosx = (1/2) sin(2x) →  

sin²x cos²x = (sinx cosx)² = [(1/2) sin(2x)]² = (1/4)sin²(2x)  

Thus the given integral becomes:  

∫ sin²x cos²x dx = ∫ (1/4)sin²(2x) dx = (1/4) ∫ sin²(2x) dx  

Now you can reduce the order of the integrand using the half-angle identity:  

sin²x = (1/2) [1 - cos(2x)]  

and Therefore:  

sin²(2x) = (1/2) [1 - cos(4x)]  

Yielding:  

(1/4) ∫ (1/2) [1 - cos(4x)] dx =  

(1/4)(1/2) ∫ [1 - cos(4x)] dx =  

(1/8) ∫ [1 - cos(4x)] dx =  

Split it into:  

(1/8) ∫ dx - (1/8) ∫ cos(4x) dx =  

(1/8)x - (1/8) (1/4)sin(4x) + C =  

(1/8)x - (1/32)sin(4x) + C  

Then, taking it back in terms of sinx, cosx, you get (recall the double-angle identity cos(2x) = cos²x - sin²x, as well):  

∫ sin²x cos²x dx = (1/8)x - (1/32)sin(4x) + C →  

∫ sin²x cos²x dx = (1/8)x - (1/32) 2sin(2x) cos(2x) + C →  

∫ sin²x cos²x dx = (1/8)x - (1/16) 2sinx cosx (cos²x - sin²x) + C →  

∫ sin²x cos²x dx = (1/8)x - (1/8) (sinx cos³x - sin³x cosx) + C →  

Finally,  

∫ sin²x cos²x dx = (1/8)x - (1/8) sinx cos³x + (1/8) sin³x cosx + C