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00:00 - 00:59 | hello friends trip to solve this question of integration in which we have to indicate the expression X and was all right of this type of questions can be solved by using integration by parts integration by parts and retainer integration of you vdx what are you and water Wii u is basically a function of X and Y is also function of x if an expression for the function is given in the form that it is a product of two functions of acts like a vehicle in the question can be solved by using we assign this as first function and dysfunction and second Express function and it can be dinner cost function as it is then we take X the integration of 1 and function - of integration of s function X the difference |

01:00 - 01:59 | first function this entire terms integration with respect to X this is what is integration by parts alright he going to his use integration by parts for this question because it is no direct formula for this particular expression of function so what is the first function and what is the second function in order to decide that you have to find out that which of the function is easier to deal with or without the function can be easily integrated sample you can easily integrate x holiday so this should be S function Sin inverse X is something which cannot be easily integrated or lead you do not know the direct it therefore we assign it at first no using integration by parts I can write a test cost function as it is that a Sin inverse X into integration of s function integration of Texas at |

02:00 - 02:59 | karuvattu into the first function which is Sin inverse x minus 1 integration is integration of the second function X differentiation differentiation of Sin inverse X with respect to X 1 by under root of 1 - x square all it takes is the expression now be simplified I can i l x square y 2 Sin inverse x minus of 1 by 2 x square under root 1 - x square x + net -2 holiday I am doing this because we do not know what direct integration formula for this particular integral now I am going to assign this integral l i |

03:00 - 03:59 | India going to find out the value of I and then put it in this particular equation so if I am so sorry for AI - of x square under root of 1 - x squared DX now it can be it can be written adding one in the unit and subtracting management it is 1 - x square minus 1 divided by under root 1 - x square DX and integration of the holiday so now it can be tailor integration 1 - x square divided by under root 1 - x square - 2 x divided by under root 1 - x square of the simplified |

04:00 - 04:59 | under root 1 - x square - 2 x divided by under root 1 - x square alright so we not direct integration formula for this holiday integration of 1 by under root 1 - x square Sin inverse X this is a difficult formula all the way to have a correct formula for this for many who do not remember it so latest record formula if we are given under root of a square minus x square and integrated with respect to its then the formula is X by 2 under root of a square minus x square + of his way to find inverse of X upon 1 |

05:00 - 05:59 | + Accountant of integration so in this particular scenario is equal to so I can be dene integrating it I am putting the value of a is equal to one in this expression for this path and it becomes X upon 2 under root 1 - X square + 1 is vital that is 1 by 2 and inverse X upon a time in vertex upon 1 + Sin inverse X + c minus the integration this part that is Sin inverse of X writing I is equals to X by 2 under root 1 - x square minus |

06:00 - 06:59 | half and inverse X + c no I know the value of this I I know the value of I am putting it in this expression so what does the initial integration because I Nishan integration cos x Sin inverse x dx it becomes let me down the top with x square by 2 Sin inverse x x square by 2 and inverse X + half of my so I have to multiply half in this product science solution and it become xy4 under root 1 - x square minus 1 by 4 Sin inverse X |

07:00 - 07:59 | + 2 is a constant of integration SO2 is just another concert I am writing that constant fever and this is going to be a final solution thank you for watching the video |

**A function `phi(x)` is called a primitive of `f(x)`; if `phi'(x) = f(x)`**

**Some important formulas of integration**

**Examples of integration: (i) `x^4` (ii) `3^x`**

**Theorem: `d/dx(int f(x) dx) = f(x)`**

**The integral of the product of a constant and a function = the constant x integral of function**

**`int {f(x) pm g(x)} dx = int f(x) dx pm int g(x) dx`**

**Geometrical interpretation of indefinite integral**

**Comparison between differentiation and integration**

**By substitution: Theorem: If `int f(x) dx = phi(x)` then `int f(ax+b) dx = 1/a phi(ax + b)dx`**

**Examples: `1/ (cos3x+1) dx` and `1/((sqrt (x+a) + sqrt (x+b))) dx`**