Advertisements
Advertisements
Question
Advertisements
Solution
\[\int\sec x \cdot \log \left( \text{sec x} + \text{tan x} \right) dx\]
\[ \text{Let log} \left( \sec x + \tan x \right) = t\]
\[ \Rightarrow \frac{\left( \sec x \tan x + \sec^2 x \right)}{\left( \sec x + \tan x \right)} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{\sec x \left( \sec x + \tan x \right)}{\left( \sec x + \tan x \right)} dx = dt\]
\[Now, \int\sec x \cdot \text{log }\left( \sec x + \tan x \right) dx\]
\[ = \ ∫ t . dt\]
\[ = \frac{t^2}{2} + C\]
\[ = \frac{\left[ \text{log} \left( \text{sec x} + \tan x \right) \right]^2}{2} + C\]
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 following definite integrals as limit of sums.
`int_1^4 (x^2 - x) dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/4) (sin x + cos x)/(9+16sin 2x) dx`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_0^1 xe^x dx = 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`
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
Evaluate the following integral:
Evaluate the following integrals as limit of sums:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` 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 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^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
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
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
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 ______.
`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 ______.
