Advertisements
Advertisements
Question
Evaluate : `int_0^(pi/2) (1)/(5 + 4 cos x)*dx`
Solution
Let I = `int_0^(pi/2) (1)/(5 + 4 cos x)*dx`
Put `tan(x/2)` = t
∴ x = 2 tan–1 t
∴ dx = `(2)/(1 + t^2)dt`
and
cos x = `(1 - t^2)/(1 + t^2)`
When x = 0, t = 0
When x = `pi/2, t = 1`
∴ I = `((2)/(1 + t^2))/(5 + 4((1 - t^2)/(1 + t^2))dt`
= `int_0^1 (2dt)/(5(1 + t^2) + 4(1 - t^2)dt`
= `int _0^1 2/(5 + 5t^2 + 4 - 4t^2)dt`
= `int_0^1 (2)/(t^2 + 9)*dt`
= `2 int_0^1 1/(t^2 + 3^2)dt`
= `2[1/3 tan^-1 t/3]_0^1`
= `2/3 [tan^(-1) 1/3 - tan^(-1) (0)]`
= `2/3 [tan^(-1) 1/3 - 0]`
= `(2)/(3) tan^-1 (1/3)`.
APPEARS IN
RELATED QUESTIONS
Evaluate:
`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`
Evaluate : `int_0^(pi/4) sin^4x*dx`
Evaluate : `int_0^(pi//4) (sin2x)/(sin^4x + cos^4x)*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Evaluate the following : `int_0^pi x sin x cos^2x*dx`
Choose the correct option from the given alternatives :
If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then
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
Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`
Evaluate the following : `int_0^(pi/2) 1/(6 - cosx)*dx`
Evaluate the following definite integral:
`int_4^9 (1)/sqrt(x)*dx`
Evaluate the following definite integrals: `int_1^2 dx/(x^2 + 6x + 5)`
Choose the correct alternative :
`int_4^9 dx/sqrt(x)` =
Choose the correct alternative :
If `int_0^"a" 3x^2*dx` = 8, then a = ?
Fill in the blank : `int_(-2)^3 dx/(x + 5)` = _______
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_(-"b")^(-"a") f(x)*dx`
State whether the following is True or False : `int_0^"a" f(x)*dx = int_"a"^0 f("a" - x)*dx`
Solve the following:
`int_0^1 e^(x^2)*x^3dx`
Solve the following : `int_2^3 x/(x^2 - 1)*dx`
Solve the following : `int_3^5 dx/(sqrt(x + 4) + sqrt(x - 2)`
Solve the following : `int_1^2 x^2*dx`
Solve the following : `int_2^4 x/(x^2 + 1)*dx`
Solve the following : `int_0^9 (1)/(1 + sqrt(x))*dx`
Choose the correct alternative:
`int_4^9 ("d"x)/sqrt(x)` =
`int_1^2 x^2 "d"x` = ______
State whether the following statement is True or False:
`int_0^1 1/(2x + 5) "d"x = log(7/5)`
`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = ______.
`int_(-2)^2 sqrt((2 - x)/(2 + x))` = ?
`int_(-5)^5 log ((7 - x)/(7 + x))`dx = ?
Evaluate the following definite integrals:
`int _1^2 (3x) / ( (9 x^2 - 1)) * dx`
Evaluate the following definite integrals: `int_-2^3 1/(x + 5) *dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5dx`
Evaluate the following definite integral:
`int_4^9 1/sqrt(x)dx`
`int_0^4 1/sqrt(4x - x^2)dx` = ______.
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5).dx`
