Advertisements
Advertisements
प्रश्न
Solve the following : `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
उत्तर
Let I = `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
= `int_0^4 (1)/sqrt(x^2 + 2x + 1 - 1 + 3)*dx`
= `int_0^4 (1)/((sqrt(x + 1))^2 + 2)*dx`
= `int_0^4 (1)/sqrt((x + 1)^2 + (sqrt(2))^2)*dx`
= `[log |x + 1 + sqrt((x + 1)62 + (sqrt(2))^2|]_0^4`
= `log |5 + sqrt(27)| - log| 1 + sqrt(3)|`
= `log |5 + 3sqrt(3)| - log| 1 + sqrt(3)|`
∴ I = `log |(5 + 3sqrt(3))/(1 + sqrt(3))|`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_0^4 (1)/sqrt(4x - x^2)*dx`
Evaluate : `int_0^(pi//4) (sin2x)/(sin^4x + cos^4x)*dx`
Evaluate : `int_1^3 (cos(logx))/x*dx`
Evaluate the following:
`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Evaluate the following : `int_0^(pi/2) 1/(6 - cosx)*dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following : `int_0^pi (sin^-1x + cos^-1x)^3 sin^3x*dx`
Evaluate the following : If f(x) = a + bx + cx2, show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`
Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`
Fill in the blank : `int_0^1 dx/(2x + 5)` = _______
State whether the following is True or False : `int_0^"a" f(x)*dx = int_"a"^0 f("a" - x)*dx`
State whether the following is True or False : `int_1^2 sqrt(x)/(sqrt(3 - x) + sqrt(x))*dx = (1)/(2)`
Solve the following:
`int_0^1 e^(x^2)*x^3dx`
`int_0^(pi/2) (cos x)/((4 + sin x)(3 + sin x))`dx = ?
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 intergral:
`int_1^2 (3x)/((9x^2 - 1))dx`
Evaluate the following definite integral :
`int_1^2 (3"x")/((9"x"^2 - 1)) "dx"`
`int_0^4 1/sqrt(4x - x^2)dx` = ______.
Solve the following.
`int_1^3 x^2 log x dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1))*dx`
