HSC Commerce कक्षा १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

Using second fundamental theorem, evaluate the following:

`int_1^"e" ("d"x)/(x(1 + logx)^3`

Using second fundamental theorem, evaluate the following:

`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7)  "d"x`

Using second fundamental theorem, evaluate the following:

`int_0^(pi/2) sqrt(1 + cos x)  "d"x`

Using second fundamental theorem, evaluate the following:

`int_1^2 (x - 1)/x^2  "d"x`

Evaluate the following:

`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`

Evaluate the following:

`int_0^2 "f"(x)  "d"x` where f(x) = `{{:(3 - 2x - x^2",", x ≤ 1),(x^2 + 2x - 3",", 1 < x ≤ 2):}`

Evaluate the following:

`int_(-1)^1 "f"(x)  "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x  < 0):}`

Evaluate the following:

f(x) = `{{:("c"x",", 0 < x < 1),(0",",  "otherwise"):}` Find 'c" if `int_0^1 "f"(x)  "d"x` = 2

Evaluate the following using properties of definite integral:

`int_(- pi/4)^(pi/4) x^3 cos^3 x  "d"x`

Evaluate the following using properties of definite integral:

`int_(- pi/2)^(pi/2) sin^2theta  "d"theta`

Evaluate the following using properties of definite integral:

`int_(-1)^1 log ((2 - x)/(2 + x))  "d"x`

Evaluate the following using properties of definite integral:

`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x)  "d"x`

Evaluate the following using properties of definite integral:

`int_0^1 log (1/x - 1)  "d"x`

Evaluate the following using properties of definite integral:

`int_0^1 x/((1 - x)^(3/4))  "d"x`

Evaluate the following:

Γ(4)

Evaluate the following:

`Γ (9/2)`

Evaluate the following:

`int_0^oo "e"^(-mx) x^6 "d"x`

Evaluate the following:

`int_0^oo "e"^(-4x) x^4  "d"x`

Evaluate the following:

`int_0^oo "e"^(- x/2) x^5  "d"x`

If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`