Advertisements
Advertisements
प्रश्न
Evaluate each of the following integral:
Advertisements
उत्तर
\[\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx\]
\[ = - \int_0^\frac{\pi}{2} e^x \left[ \cos x + \left( - \sin x \right) \right]dx\]
\[ = \left.- {e^x \cos x}\right|_0^\frac{\pi}{2} .............\left\{ \int e^x \left[ f\left( x \right) + f'\left( x \right) \right]dx = e^x f\left( x \right) + C \right\}\]
\[ = - \left( e^\frac{\pi}{2} \cos\frac{\pi}{2} - e^0 \cos0 \right)\]
\[ = - \left( e^\frac{\pi}{2} \times 0 - 1 \times 1 \right)\]
\[ = - \left( 0 - 1 \right)\]
\[ = 1\]
संबंधित प्रश्न
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
\[\int\limits_0^1 \log\left( 1 + x \right) dx\]
\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx\]
\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
Evaluate `int "dx"/sqrt((x - alpha)(beta - x)), beta > alpha`
If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.
What is an antiderivative of \[2x+1\]?
