Advertisements
Advertisements
Question
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
Options
`(- 1)/(sin x + cos x) + "C"`
log |sin x + cos x| + C
log |sin x - cos x| + C
`1/(sin x + cos x)^2`
Solution
`int (cos 2x)/(sin x + cos x)^2dx` is equal to log |sin x + cos x| + C.
Explanation:
Let `I = (cos 2x)/(sin x + cos x) dx`
`= int (cos^2 x - sin^2 x)/(cos x + sin x)^2 dx`
`= int ((cos x - sin x)(cos x + sin x))/(cos x + sin x)^2 dx`
`= int (cos x - sin x)/(cos x + sin x) dx`
put cos x + sin x = t
⇒ (-sin x + cos x)dx = dt
`= int dt/t = log t + C`
= log |sin x + cos x| + C
APPEARS IN
RELATED QUESTIONS
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Prove the following:
`int_0^1 xe^x dx = 1`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
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`
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
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 ______.
