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`y=int_(0)^(x)|t|dt` <br> Case I: If `xgt0` <br> `y=int_(0)^(x)tdt=[(t^(2))/2]_(0)^(x)=(x^(2))/2implies(dy)/(dx)=x=2` (given) <br> `implies x=2` and `y=2` <br> `:.` eqution of tangent is `(y-2)=2(x-2)` <br> or `y-2x+2=0` <br> Hence `x` intercept `=` <br> Case II: `xlt0` <br> `-y=int_(0)^(pi)-tdt=[(-t^(2))/2]_(0)^(x)=(-x^(2))/2` <br> `:.(dy)/(dt)=-x=2` <br> `:. x=-2,:.y=-2`, <br> `:.` equation iof tangent is `y+2=2(x+2)` <br> or `2x-y+2=0` <br> `:.x` intencept `=-1`**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`**