Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
Advertisements
उत्तर
Let `I = int_0^2 x sqrt (2 - x) dx`
Put 2 - x = t
⇒ dx = dt
When x = 0, t = 2
and x = 2, t = 0
∵ `I = - int_2^0 (2 - t) sqrtt dt`
`= int_0^2 (2t^(1/2) - t^(3/2)) dt`
`= [(2t^(3/2))/(3/2) - t^(5/2)/(5/2)]_0^2` `...[∵ - int_a^0 f (x) dx = int_0^a f (x) dx]`
`= [4/3 t^(3/2) - 2/5 t^(5/2)]_0^2`
`= 4/3 (2)^(3/2) - 2/5 (2)^(5/2)`
`= 4/3 xx 2 sqrt2 - 2/5 xx 4 sqrt2`
`= (8sqrt2)/3 - (8 sqrt 2)/5`
`= (16 sqrt2)/15`
APPEARS IN
संबंधित प्रश्न
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : ∫ log (1 + x2) dx
`int_0^2 e^x dx` = ______.
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_0^1 x(1 - x)^5 "d"x`
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_4^9 1/sqrt(x)dx` = ______.
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate the following definite integral:
`int_4^9 1/sqrt"x" "dx"`
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
