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00:00 - 00:59 | hello students in this question we need to find the integration of e raise to power x divided by X in bracket 1 + X log x dx we need to find the integration of this so we can write as I equal to integration of e raise to power x divided by X in bracket 1 + X log x dx now first of four foot X inside the bracket so we can write as I was to integration of e raise to power x 1 by X + X log x divided by x dx so this X and this access cancelled out so we can write as I equals to integration of e raise to power x 1 by X + log x dx now we know that integration of e raise to power x f of X + |

01:00 - 01:59 | x dx equal to raise to power xfx and in this part we can say that log let support log x equal to log x equal to so we can write as one by x dx equal to DT if our function f of X equal to log x then h f 10 X equal to 1 by X so from here so from this formula we can write as integration equals to e raise to power xfx so power FX equal to log x power FX equal to log x integration equal to e raise to power x log x to this is the value of integration 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`**