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Text Solution

`"Ln "|x-sqrt(x^(2)-1)|-tan^(-1)x+c``"In "|x+sqrt(x^(2)-1)|-tan^(-1)x+c``"In "|x-sqrt(x^(2)-1)|-sec^(-1)x+c``"In "|x+sqrt(x^(2)-1)|-sec^(-1)x+c`

Answer :

DSolution :

`I=int(sqrt(x-1))/(xsqrt(x+1))dx=int(x-1)/(xsqrt(x^(2)-1))dx` <br> `=int(dx)/(sqrt(x^(2)-1))-int(dx)/(xsqrt(x^(2)-1))` <br> `="In " |x+sqrt(x^(2)-1)|-sec^(-1)x+c`**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`**