Advertisements
Advertisements
Question
Evaluate each of the following integral:
Advertisements
Solution
\[\int_0^\frac{\pi}{4} \tan xdx\]
\[ = \left.{\log\sec\ x}\right|_0^\frac{\pi}{4} \]
\[ = \log\sec\frac{\pi}{4} - \log\sec0\]
\[ = \log\sqrt{2} - \log1\]
\[ = \log 2^\frac{1}{2} - 0\]
\[ = \frac{1}{2}\log2\]
APPEARS IN
RELATED QUESTIONS
Evaluate the following integral:
Evaluate each of the following integral:
\[\int_a^b \frac{x^\frac{1}{n}}{x^\frac{1}{n} + \left( a + b - x \right)^\frac{1}{n}}dx, n \in N, n \geq 2\]
Evaluate the following integral:
If f(x) is a continuous function defined on [−a, a], then prove that
Prove that:
\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals
If \[\int\limits_0^a \frac{1}{1 + 4 x^2} dx = \frac{\pi}{8},\] then a equals
The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
\[\int\limits_0^1 \tan^{- 1} x dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_1^2 \frac{1}{x^2} e^{- 1/x} dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
\[\int\limits_0^\pi \frac{dx}{6 - \cos x}dx\]
\[\int\limits_2^3 e^{- x} dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
Evaluate the following:
`int_0^oo "e"^(- x/2) x^5 "d"x`
If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`
Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`
Evaluate the following:
`int ((x^2 + 2))/(x + 1) "d"x`
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
