Advertisements
Advertisements
प्रश्न
Evaluate the following integrals:
`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`
उत्तर
Let I = `int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx` ...(i)
= `int_1^3 (root(3)((1 + 3 - x) + 5))/(root(3)((1 + 3 - x) + 5) + root(3)(9 - (1 + 3 - x)))*dx ...[because int_"a"^"b" f(x)*dx = int_"a"^"b" f("a" + "b" - x)*dx]`
∴ I = `int_1^3 (root(3)(9 - x))/(root(3)(9 - x) + root(3)(5 + x))*dx` ...(ii)
Adding (i) and (ii), we get
2I = `int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx + int_1^3 (root(3)(9 - x))/(root(3)(9 - x) + root(3)(5 + x))*dx`
= `int_1^3 (root(3)(x + 5) + root(3)(9 - x))/(root(3)(x + 5) - root(3)(9 - x))*dx`
= `int_1^3 1*dx`
= `[x]_1^3`
∴ 2I = 3 – 1 = 2
∴ I = 1
APPEARS IN
संबंधित प्रश्न
Prove that:
`{:(int_(-a)^a f(x) dx = 2 int_0^a f(x) dx",", "If" f(x) "is an even function"),( = 0",", "if" f(x) "is an odd function"):}`
Show that: `int _0^(pi/4) log (1 + tanx) dx = pi/8 log2`
Evaluate : `int_0^4 (1)/sqrt(4x - x^2)*dx`
Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*dx`
Evaluate : `int_1^3 (cos(logx))/x*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
Evaluate the following : `int_0^1 (log(x + 1))/(x^2 + 1)*dx`
`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = ______.
Choose the correct option from the given alternatives :
If `dx/(sqrt(1 + x) - sqrt(x)) = k/(3)`, then k is equal to
Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))*dx`
Choose the correct alternative :
`int_4^9 dx/sqrt(x)` =
Choose the correct alternative :
`int_2^3 x^4*dx` =
Choose the correct alternative :
`int_(-7)^7 x^3/(x^2 + 7)*dx` =
Fill in the blank : `int_0^2 e^x*dx` = ________
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_(-"b")^(-"a") f(x)*dx`
Solve the following : `int_1^2 x^2*dx`
Solve the following : `int_1^2 dx/(x(1 + logx)^2`
State whether the following statement is True or False:
`int_0^1 1/(2x + 5) "d"x = log(7/5)`
Evaluate the following definite integral:
`int_4^9 1/sqrtx dx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int_0^1 |x| dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
Solve the following.
`int_1^3 x^2 log x dx`
The principle solutions of the equation cos θ = `1/2` are ______.
Evaluate the following definite intergral:
`int_4^9 1/sqrtx dx`
Evaluate the following definite intergral:
`int_1^2(3x)/(9x^2-1).dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
Evaluate the following definite intergral:
`int_(-2)^3 1/(x + 5)dx`
