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Answer :

`I=(1)/(6(m+1)){2x^(3m)+3x^(2m)+6x^(m)}^((m+1)/(m))+C`Transcript

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00:00 - 00:59 | hello everyone today question it for any natural number and evaluate integration extra part 3 and + Exeter part 2 and + x to the power x into 2 Episode 52 in the current plastic phone to the power 1 by DX so that start with the integration given this time we can write this extra class part 1 + x to the power M / Into X into two x ^ 2 m + 3 X ^ M power 1 by ok now now this one it and write it integration X ^ 3 - 1 + X ^ 2 and minus 1 + x to the power minus 5 ok for this part is that we can write |

01:00 - 01:59 | power bi if you write like this then this will go inside the bracket and we can write this as to X ^ 3M + 3 X ^ 2 + 6 x ^ ^ 2 X ^ 3M + 3 X ^ 2 m + 6 x ^ M = 23 now differentiating we get with respect to X we get 6 m16 x 6M Into X ^ 2 and -1 + 6 m interest to the power n -1 X equals to now this weekend ride to the power 3 minus 1 + 2 x to the power to a - 1 |

02:00 - 02:59 | + x to the power n minus 1 into DX equal to 3 by 6 m6m common from all the numbers now integration of the value of this integration will be so this part is T20 access this one ok so this one in this one and the same so we can write integration of e to the power 1 by m into ti ti bi ok because it was our integration is 1 by 6 into 10 to the power 1 by M + 1 divided by 1 by N ok equals to 1 by 6 into 10 to the power 1 + MBM divided by 1 + 2 + 6 this and will get cancelled today |

03:00 - 03:59 | score the value of things we get one by X into 1 + 10 + 2 X 3M + 5 x ^ 2 and + 6 x ^ m into 1 + cos to the power 1 + 1 by X + 6 is required solution thank you |

**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`**