Advertisements
Advertisements
Question
Evaluate : `int_(-4)^2 (1)/(x^2 + 4x + 13)*dx`
Solution
`int_(-4)^2 (1)/(x^2 + 4x + 13)*dx`
= `int_(-4)^2 (1)/(x^2 + 4x + 4 + 9)*dx`
= `int_(-4)^2 (1)/((x + 2)^2 + 3^2)*dx`
= `[1/3tan^-1 ((x + 2)/3)]_(-4)^2`
= `(1)/(3)tan^-1 ((2 + 2)/3) - (1)/(3)tan^-1((-4 + 2)/3)`
= `(1)/(3)tan^-1 (4/3) - (1)/(3)tan^-1 (-2/3)`
= `(1)/(3)[tan^-1 4/3 + tan^-1 2/3]`. ...[∵ tan–1 (–x) = –tan–1 x]
APPEARS IN
RELATED QUESTIONS
Evaluate:
`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`
Evaluate : `int_0^(pi//4) (sin2x)/(sin^4x + cos^4x)*dx`
Evaluate:
`int_0^(pi/2) sqrt(cos x) sin^3x * dx`
Evaluate : `int_0^(pi/4) sec^4x*dx`
Evaluate : `int _((1)/(sqrt(2)))^1 (e^(cos^-1x) sin^-1x)/(sqrt(1 - x^2))*dx`
Evaluate the following : `int_0^3 x^2(3 - x)^(5/2)*dx`
Choose the correct option from the given alternatives :
If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then
Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`
Evaluate the following : `int_0^pi x*sinx*cos^4x*dx`
Evaluate the following : `int_0^(pi/2) 1/(6 - cosx)*dx`
Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`
Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`
Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`
Choose the correct alternative :
`int_4^9 dx/sqrt(x)` =
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_(-2)^3 dx/(x + 5)` = _______
State whether the following is True or False : `int_(-5)^(5) x^3/(x^2 + 7)*dx` = 0
Solve the following:
`int_1^3 x^2 log x*dx`
Solve the following : `int_4^9 (1)/sqrt(x)*dx`
Solve the following : `int_0^1 (x^2 + 3x + 2)/sqrt(x)*dx`
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"c""f"(x) "d"x + int_"c"^"b" "f"(x) "d"x`, where a < c < b
`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + 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 integrals:
`int_-9^9 (x^3)/(4 - x^2) dx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/((9x^2-1 )`dx
Solve the following.
`int_1^3 x^2 log x dx`
`int_a^b f(x) dx = int_a^b f (t) dt`
Evaluate the following definite integral:
`int_1^3 logx dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Solve the following:
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^3logxdx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Solve the following.
`int_1^3x^2 logx dx`
Solve the following.
`int_1^3 x^2 log x dx`
Solve the following.
`int_1^3x^2logx dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
Evaluate the following definite intergral.
`int_1^2 (3x)/((9x^2 - 1))dx`
