Advertisements
Advertisements
प्रश्न
Evaluate : `int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3))*dx`
उत्तर
`int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3))*dx`
= `int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3)) xx (sqrt(2x + 3) + sqrt(2x - 3))/(sqrt(2x + 3) + sqrt(2x - 3))*dx`
= `int_3^5 (sqrt(2x + 3) + sqrt(2x - 3))/((2x + 3) - (2x - 3))*dx`
= `(1)/(6) int_3^5 (2x + 3)^(1/2)*dx + (1)/(6) int_3^5 (2x - 3)^(1/2)*dx`
= `(1)/(6)[(2x + 3^(3/2))/(2(3/2))]_3^5 + (1)/(6)[((2x - 3)^(3/2))/(2(3/2))]_3^5`
= `(1)/(18)[(10 + 3)^(3/2) - (6 + 3)^(3/2)] + (1)/(18)[(10 - 3)^(3/2) - (6 - 3)^(3/2)]`
= `(1)/(18)[13sqrt(13) - 9sqrt(9)] + (1)/(18)[7sqrt(7) - 3sqrt(3)]`
= `(1)/(18)(13sqrt(13) - 27 + 7sqrt(7) - 3sqrt(3))`
= `(1)/(18)(13sqrt(13) + 7sqrt(7) - 3sqrt(3) - 27)`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int 1/(a^2 - x^2) dx = 1/2 a log ((a +x)/(a-x)) + c`
Evaluate : `int_0^1 x tan^-1x*dx`
Evaluate : `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`
Evaluate : `int_0^pi (1)/(3 + 2sinx + cosx)*dx`
Evaluate : `int_0^(pi/4) sec^4x*dx`
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Choose the correct option from the given alternatives :
Let I1 = `int_e^(e^2) dx/logx "and" "I"_2 = int_1^2 e^x/x*dx`, then
Choose the correct option from the given alternatives :
`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Evaluate the following : `int_0^1 1/(1 + sqrt(x))*dx`
Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`
Evaluate the following : `int_0^1 (cos^-1 x^2)*dx`
Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`
Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`
Evaluate the following : if `int_a^a sqrt(x)*dx = 2a int_0^(pi/2) sin^3x*dx`, find the value of `int_a^(a + 1)x*dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))*dx`
Choose the correct alternative :
`int_"a"^"b" f(x)*dx` =
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_(-9)^9 x^3/(4 - x^2)*dx` = _______
State whether the following is True or False : `int_4^7 ((11 - x)^2)/((11 - x)^2 + x^2)*dx = (3)/(2)`
Solve the following : `int_(-2)^3 (1)/(x + 5)*dx`
Solve the following : `int_1^2 x^2*dx`
Solve the following : `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
Prove that: `int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"(2"a" - x) "d"x`
State whether the following statement is True or False:
`int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"("a" - x) "d"x`
By completing the following activity, Evaluate `int_1^2 (x + 3)/(x(x + 2)) "d"x`
Solution: Let I = `int_1^2 (x + 3)/(x(x + 2)) "d"x`
Let `(x + 3)/(x(x + 2)) = "A"/x + "B"/((x + 2))`
∴ x + 3 = A(x + 2) + B.x
∴ A = `square`, B = `square`
∴ I = `int_1^2[("( )")/x + ("( )")/((x + 2))] "d"x`
∴ I = `[square log x + square log(x + 2)]_1^2`
∴ I = `square`
Prove that: `int_0^(2a) f(x)dx = int_0^a f(x)dx + int_0^a f(2a - x)dx`
Evaluate the following definite integrals:
`int _1^2 (3x) / ( (9 x^2 - 1)) * dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))dx`
`int_0^1 1/(2x + 5)dx` = ______
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Solve the following:
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_1^3x^2logx dx`
Evaluate the following definite intergral:
`\underset{4}{\overset{9}{int}}1/sqrt(x)dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
