© examsiri.com
Question : 74 of 160
Marks:
+1,
-0
Solution:
We have,
In=∫xn‌sin‌x‌d‌x ∴I6=∫x6‌sin‌x‌d‌x =x6(−cos‌x)−∫−cos‌x⋅(6x5)dx =−x6‌cos‌x+6‌∫x5‌cos‌x‌d‌x =−x6‌cos‌x+6[x5‌sin‌x−∫sin‌x⋅(5x4)dx]
=−x6‌cos‌x+6x5‌sin‌x−30‌∫x4‌sin‌x‌d‌x
=−x6‌cos‌x+6x5‌sin‌x−30[x4(−cos‌x)−∫−cos‌x(4x3)‌d‌x]
=−x6‌cos‌x+6x5‌sin‌x+30x4‌cos‌x‌‌−120‌∫x3‌cos‌x‌d‌x
=−x6‌cos‌x+6x5‌sin‌x+30x4‌cos‌x−120[x3‌sin‌x−∫sin‌x(3x2)‌d‌x]
=−x6‌cos‌x+6x5‌sin‌x+30x4‌cos‌x−120x3‌sin‌x+360‌∫x2‌sin‌x‌d‌x
=−x6‌cos‌x+6x5‌sin‌x+30x4‌cos‌x −120x3‌sin‌x+360I2
∴I6−360I2=(−x6+30x4)‌cos‌x +(6x5−120x3)‌sin‌x
∴‌‌f(x)=−x6+30x4 and g(x)=6x5−120x3
∴‌‌f(1)+g(1)=−1+30+6−120=−85
© examsiri.com
Go to Question: