Advertisements
Advertisements
Question
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Advertisements
Solution
`= int_0^(pi/2) sin^3 x dx`
`= 1/4 int_0^(pi/2) (3 sinx - sin 3x) dx` `....[∵ sin 3x = 3 sin x - 4 sin 3x]`
`= 1/4 [-3 cos x + (cos 3x)/3]_0^(pi/2)`
`= 1/4 [- 3 cos pi/2 + 1/3 cos (3pi)/2] - 1/4 [- 3 cos 0 + (cos0)/3]`
`= 1/4 [0 + 0 + 3 - 1/3]`
`= 1/4 (8/3)`
`= 2/3`
APPEARS IN
RELATED QUESTIONS
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Evaluate the definite integral:
`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
`int dx/(e^x + e^(-x))` is equal to ______.
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.
(A) `1/2`
(B) `1/3`
(C) `1/4`
(D) `1/5`
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
Evaluate the following integrals as limit of sums:
Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
What is the derivative of `f(x) = |x|` at `x` = 0?
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
