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00:00 - 00:59 | hello students in the given question for a positive integer n let iron equals to integral pi by 2 minus x x cos x dx eliminated from - 52 53 54 solution equal to pi by 2 minus Mod x x cos x dx now hair let's suppose that affects equals to pi by 2 minus Mod x x Cos NX for f of minus X equal to pi by 2 minus now hair it since it is MOD X Mod X can also be written as smart so Mod of -6 can also be written as Mod x |

01:00 - 01:59 | in place of model -6 you are writing Mod of x x know the value of cos of minus theta = 22 the value of cos theta so here when we put a call me when we put -6 in place of XP get cause of minus 10 X with can also be written as cause of NX so now that this is an even function so sensors and even function the value of this property can be used with said that when the limits of minus 8 28 f x dx we can write this as integral f x x 2 x y Limited limits are from 0 to using this property in the above in this question we write I N = 22 x integral limit from 0 to Pi |

02:00 - 02:59 | X pi by 2 minus x since the value of x from 0 to pi in place of Modicare writing x x cos x dx taking to inside the integration we get integral limit from 0 to pi by 5 - 2 x x cos x dx Nau Nau writing this opening this is brackets and writing it separately we get interior limit from 0 to pi since the constant of taking it out of the integration we get 5 x Cos NX d x minus 2 into the constants of taking out of the integration limit from 0 to buy eggs cause n x dx now the Integration of Cos of n |

03:00 - 03:59 | x dx is sin NX upon and limit from 0 to buy - now here we are we have to use integration by parts so integrating this with the help of integration by parts let's assume you equals to X and Y is equals to cos of annexed by assuming this this can be integrated and integration we have x sin x sin x upon in minus integral sin NX sin x upon in dx12 Limited from 0 to pi scan for the bitterness by X flying limit we get sign and 5 upon in -0 |

04:00 - 04:59 | since we are putting zero in place of eggs to x x sin NX upon in minus the integration of sin x upon X is nothing but cause NX upon n square and there is a minor so that becomes less well images from 0 to pi so the scan for the be written as by x sin n Pi upon and my applying limits Pi sin n pi by N + cos n pi by n square minus minus 0 - |

05:00 - 05:59 | cause Dhirubhai in square so this whole can further be written as 0 - 2 time since the value of sin and 500 - 2 X 0 + cos n Pi bi n square minus 1 by n square so it came out to be to buy n square x by 1 minus Cos of N Pi so therefore the value of iron comes out to be to buy and Square X by 1 minus Cos n Pi now therefore the value of iron + 3 + iPhone |

06:00 - 06:59 | greatest integer Rao comes out to be now placing for even placing and equal to 1 for it placing an equal to 3 and 4 by 4 placing n = 24 to this can be written as 2 X 1 - cos A + 2 by 9 X 1 - cos 3 pi by 2 + 2 by 16 x minus cos 4pi so this can cause a be written as the value of 2 X 1 - cos 5 came out to be for Plus the value of 219 X 1 - cos 35 came out to be 49 and the value of 2 by 16 X 1 - cos 4pi came out to be zero then the value of cos 5 is -1 the value of cos |

07:00 - 07:59 | 3pi is also -1 in the value of cos 4 Pi zero so applying these values Now since the greatest integer is applied so the greatest integer of all this comes out to be 24 is the final answer for this question |

**Fundamental Theorem of Definite Integration**

**`int_a ^b f(x) dx = phi(b) - phi(a)`**

**Examples: `int_2 ^4 x / (x^2 + 1) dx`**

**Definite integration by substitution**

**Examples: `int_0 ^1 sin^-1 ((2x )/ (1 + x^2)) dx`**

**Property 1: Integration is independent of the change of variable. `int_a ^b f(x) dx = int_a ^b f(t) dt`**

**Property 2: If the limits of a definite integral are interchanged then its value changes. `int_a ^b f(x) dx = - int_b ^a f(x) dx`**

**Property 3: `int_a ^b f(x) dx = int_a ^c f(x)dx + int_c ^b f(x) dx`**

**Property 4: If `f(x)` is a continuous function on `[a,b]` then `int_a ^b f(x) dx = int_a ^b f(a+b-x) dx`**

**Property 5: If `f(x)` is a continuous function defined on `[0,a]` then `int_0 ^a f(x) dx = int_0^a f(a-x) dx`**