Advertisements
Advertisements
Question
Evaluate:
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
Solution
`int_0^(pi/4) sqrt(1 + sin 2x)*dx`
= `int_0^(pi/4) sqrt(sin^2x + cos^2x + 2 sin x cos x)*dx`
= `int_0^(pi/4) sqrt((sinx + cosx)^2)*dx`
= `int_0^(pi/4) (sinx + cosx)*dx`
= `int_0^(pi/4) sinx*dx + int_0^(pi/4) cosx*dx`
= `[ - cos x]_0^(pi/4) + [sin x]_0^(pi/4)`
= `[- cos pi/4 - (- cos 0)] + [sin pi/4 - sin 0]`
= `-(1)/sqrt(2) + 1 + (1)/sqrt(2) - 0`
= 1.
APPEARS IN
RELATED QUESTIONS
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"):}`
Evaluate : `int_0^(pi/4) sin 4x sin 3x *dx`
Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*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`
Choose the correct option from the given alternatives :
`int_1^2 (1)/x^2 e^(1/x)*dx` =
Evaluate the following : `int_(pi/4)^(pi/2) (cos theta)/[cos theta/2 + sin theta/2]^3*d theta`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following definite integrals: `int_2^3 x/((x + 2)(x + 3)). dx`
Evaluate the following integrals:
`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_0^1 dx/(2x + 5)` = _______
Fill in the blank : If `int_0^"a" 3x^2*dx` = 8, then a = _______
Solve the following : `int_3^5 dx/(sqrt(x + 4) + sqrt(x - 2)`
Solve the following : `int_1^2 x^2*dx`
`int_1^9 (x + 1)/sqrt(x) "d"x` =
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
Choose the correct alternative:
`int_2^3 x^4 "d"x` =
State whether the following statement is True or False:
`int_0^1 1/(2x + 5) "d"x = log(7/5)`
Evaluate `int_1^2 (3x)/((9x^2 - 1)) "d"x`
Evaluate `int_1^2 "e"^(2x) (1/x - 1/(2x^2)) "d"x`
Evaluate `int_1^3 log x "d"x`
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
`int_2^3 "x"/("x"^2 - 1)` dx = ____________.
Evaluate the following definite integrats:
`int_4^9 1/sqrt x dx`
Evaluate the following definite integral :
`int_1^2 (3"x")/((9"x"^2 - 1)) "dx"`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrt(x)dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5) *dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Solve the following.
`int_0 ^1 e^(x^2) * x^3`dx
Evaluate the following definite intergral:
`int_4^9 1/sqrtx dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
Evaluate the following definite intergral:
`\underset{4}{\overset{9}{int}}1/sqrt(x)dx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5)dx`
Evaluate the following definite intergral:
`int_4^9(1)/sqrtxdx`
