Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 2 - Integral Calculus – 1 [Latest edition]

Integrate the following with respect to x.

`sqrt(3x + 5)`

Integrate the following with respect to x.

`(9x^2 - 4/x^2)^2`

Integrate the following with respect to x.

(3 + x)(2 – 5x)

Integrate the following with respect to x.

`sqrt(x)(x^3 - 2x + 3)`

Integrate the following with respect to x.

`(8x + 13)/sqrt(4x + 7)`

Integrate the following with respect to x.

`1/(sqrt(x + 1) + sqrt(x - 1))`

Integrate the following with respect to x.

If f'(x) = x + b, f(1) = 5 and f(2) = 13, then find f(x)

Integrate the following with respect to x.

If f'(x) = 8x³ – 2x and f(2) = 8, then find f(x)

Integrate the following with respect to x.

`(sqrt(2x) - 1/sqrt(2x))^2`

Integrate the following with respect to x.

`(x^4 - x^2 + 2)/(x - 1)`

Integrate the following with respect to x.

`x^3/(x + 2)`

Integrate the following with respect to x.

`(x^3 + 3x^2 - 7x + 11)/(x + 5)`

Integrate the following with respect to x.

`(3x + 2)/((x - 2)(x - 3))`

Integrate the following with respect to x.

`(4x^2 + 2x + 6)/((x + 1)^2(x - 3))`

Integrate the following with respect to x.

`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))`

Integrate the following with respect to x.

If f'(x) = `1/x` and f(1) = `pi/4`, then find f(x)

Integrate the following with respect to x.

`"e"^(xlog"a") + "e"^("a"log"a") - "e"^("n"logx)`

Integrate the following with respect to x.

`("a"^x - "e"^(xlog"b"))/("e"^(x log "a") "b"^x)`

Integrate the following with respect to x.

`("e"^x + 1)^2 "e"^x`

Integrate the following with respect to x.

`("e"^(3x) - "e"^(-3x))/"e"^x`

Integrate the following with respect to x.

`("e"^(3x) +"e"^(5x))/("e"^x + "e"^-x)`

Integrate the following with respect to x.

`[1 - 1/2]"e"^((x + 1/x))`

Integrate the following with respect to x.

`1/(x(log x)^2`

Integrate the following with respect to x.

If f'(x) = e^x and f(0) = 2, then find f(x)

Integrate the following with respect to x.

2 cos x – 3 sin x + 4 sec²x – 5 cosec²x

Integrate the following with respect to x.

sin³x

Integrate the following with respect to x.

`(cos 2x + 2sin^2x)/(cos^2x)`

Integrate the following with respect to x.

`1/(sin^2x cos^2x) ["Hint:" sin^2x + cos^2x = 1]`

Integrate the following with respect to x.

`sqrt(1 - sin 2x)`

Integrate the following with respect to x.

xe^–x 

Integrate the following with respect to x.

x³e^3x

Integrate the following with respect to x.

log x

Integrate the following with respect to x.

x log x

Integrate the following with respect to x.

xⁿ log x

Integrate the following with respect to x.

`x^5  "e"^x`

Integrate the following with respect to x.

`(2x + 5)/(x^2 + 5x - 7)`

Integrate the following with respect to x.

`("e"^(3logx))/(x^4 + 1)`

Integrate the following with respect to x.

`"e"^(2x)/("e"^(2x) - 2)`

Integrate the following with respect to x.

`(log x)^3/x`

Integrate the following with respect to x.

`(6x + 7)/sqrt(3x^2 + 7x - 1)`

Integrate the following with respect to x.

`(4x + 2) sqrt(x^2 + x + 1)`

Integrate the following with respect to x.

x⁸(1 + x⁹)⁵ 

Integrate the following with respect to x.

`(x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x)`

Integrate the following with respect to x

`1/(x log x)`

Integrate the following with respect to x.

`x/(2x^4 - 3x^2 - 2)`

Integrate the following with respect to x.

e^x(1 + x) log(xe^x)

Integrate the following with respect to x.

`1/(x^2(x^2 + 1))`

Integrate the following with respect to x.

`"e"^x [1/x^2 - 2/x^3]`

Integrate the following with respect to x.

`"e"^x [(x - 1)/(x + 1)^3]`

Integrate the following with respect to x.

`"e"^(3x) [(3x - 1)/(9x^2)]`

Integrate the following with respect to x.

`1/(9 - 16x^2)`

Integrate the following with respect to x.

`1/(9 - 8x - x^2)`

Integrate the following with respect to x.

`1/(2x^2 - 9)`

Integrate the following with respect to x.

`1/(x^2 - x - 2)`

Integrate the following with respect to x.

`1/(x^2 + 3x + 2)`

Integrate the following with respect to x.

`1/(2x^2 + 6x - 8)`

Integrate the following with respect to x.

`"e"^x/("e"^(2x) - 9)`

Integrate the following with respect to x.

`1/sqrt(9x^2 - 7)`

Integrate the following with respect to x.

`1/sqrt(x^2 + 6x + 13)`

Integrate the following with respect to x.

`1/sqrt(x^2 - 3x + 2)`

Integrate the following with respect to x.

`x^3/sqrt(x^8 - 1)`

Integrate the following with respect to x.

`sqrt(1 + x + x^2)`

Integrate the following with respect to x.

`sqrt(x^2 - 2)`

Integrate the following with respect to x.

`sqrt(4x^2 - 5)`

Integrate the following with respect to x.

`sqrt(2x^2 + 4x + 1)`

Integrate the following with respect to x.

`1/(x + sqrt(x^2 - 1)`

Using second fundamental theorem, evaluate the following:

`int_0^1 "e"^(2x)  "d"x`

Using second fundamental theorem, evaluate the following:

`int_0^(1/4) sqrt(1 - 4)  "d"x`

Using second fundamental theorem, evaluate the following:

`int_1^2 (x "d"x)/(x^2 + 1)`

Using second fundamental theorem, evaluate the following:

`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`

Using second fundamental theorem, evaluate the following:

`int_0^1 x"e"^(x^2)  "d"x`

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`

Evaluate the following integrals as the limit of the sum:

`int_0^1 (x + 4)  "d"x`

Evaluate the following integrals as the limit of the sum:

`int_1^3 x  "d"x`

Evaluate the following integrals as the limit of the sum:

`int_1^3 (2x + 3)  "d"x`

Evaluate the following integrals as the limit of the sum:

`int_0^1 x^2  "d"x`

Choose the correct alternative:

`int 1/x^3  "d"x` is

Choose the correct alternative:

`int 2^x  "d"x` is

Choose the correct alternative:

`int (sin2x)/(2sinx)  "d"x` is

Choose the correct alternative:

`int (sin5x - sinx)/(cos3x)  "d"x` is

Choose the correct alternative:

`int logx/x  "d"x, x > 0` is

Choose the correct alternative:

`int "e"^x/sqrt(1 + "e"^x)  "d"x` is

Choose the correct alternative:

`int sqrt("e"^x) "d"x` is

Choose the correct alternative:

`int "e"^(2x) [2x^2 + 2x]  "d"x`

Choose the correct alternative:

`int "e"^x/("e"^x + 1)  "d"x` is

Choose the correct alternative:

`int[9/(x - 3) - 1/(x + 1)]  "d"x` is

Choose the correct alternative:

`int (2x^3)/(4 + x^4)  "d"x` is

Choose the correct alternative:

`int ("d"x)/sqrt(x^2 - 36) + "c"`

Choose the correct alternative:

`int (2x + 3)/sqrt(x^2 + 3x + 2)  "d"x` is

Choose the correct alternative:

`int_0^1 (2x + 1)  "d"x` is

Choose the correct alternative:

`int_2^4 ("d"x)/x` is

Choose the correct alternative:

`int_0^oo "e"^(-2x)  "d"x` is

Choose the correct alternative:

`int_(-1)^1 x^3 "e"^(x^4)  "d"x` is

Choose the correct alternative:

If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x)  "d"x + int_"c"^"b" f(x)  "d"x` is

Choose the correct alternative:

The value of `int_(- pi/2)^(pi/2) cos  x  "d"x` is

Choose the correct alternative:

`int_0^1 sqrt(x^4 (1 - x)^2)  "d"x` is

Choose the correct alternative:

If `int_0^1 f(x)  "d"x = 1, int_0^1 x  f(x)  "d"x = "a"`, and `int_0^1 x^2 f(x)  "d"x = "a"^2`, then `int_0^1 ("a" - x)^2 f(x)  "d"x` is

Choose the correct alternative:

The value of `int_2^3 f(5 - 3)  "d"x - int_2^3 f(x)  "d"x` is

Choose the correct alternative:

`int_0^4 (sqrt(x) + 1/sqrt(x))  "d"x` is

Choose the correct alternative:

`int_0^(pi/3) tan x  "d"x` is

Choose the correct alternative:

Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is

Choose the correct alternative:

Γ(n) is

Choose the correct alternative:

Γ(1) is

Choose the correct alternative:

If n > 0, then Γ(n) is

Choose the correct alternative:

`Γ(3/2)`

Choose the correct alternative:

`int_0^oo x^4"e"^-x  "d"x` is

Evaluate the following integral:

`int 1/(sqrt(x + 2) - sqrt(x + 3))  "d"x`

Evaluate the following integral:

`int ("d"x)/(2 - 3x - 2x^2)`

Evaluate the following integral:

`int ("d"x)/("e"^x + 6 + 5"e"^-x)`

Evaluate the following integral:

`int sqrt(2x^2 - 3)  "d"x`

Evaluate the following integral:

`sqrt(9x^2 + 12x + 3)  "d"x`

Evaluate the following integral:

`int (x + 1)^2 log x  "d"x`

Evaluate the following integral:

`int log (x - sqrt(x^2 - 1)) "d"x`

Evaluate the following integral:

`int_0^1 sqrt(x(x - 1))  "d"x`

Evaluate the following integral:

`int_(-1)^1 x^2 "e"^(-2x)  "d"x`

Evaluate the following integral:

`int_0^3 (x dx)/(sqrt(x + 1)+ sqrt(5x + 1))`