$$ \int \! u \ dv = uv \ – \int \! v \ du$$
\( \int \! \ln x \ dx \)
\( u = \ln x \ \ \ \ \ \ du = \frac{1}{x} \ dx \)
\( dv = dx \ \ \ \ \ \ v = x \)
\( \int \! \ln x \ dx = x \ln x – \int \! x \frac{1}{x} \ dx \)
\( \int \! \ln x \ dx = x \ln x – x + C \)
\( \int \! x \ln x \ dx \)
\( u = \ln x \ \ \ \ \ \ du = \frac{1}{x} \ dx \)
\( dv = x \ dx \ \ \ \ \ \ v = \frac{x^2}{2}\)
\( \int \! x \ln x \ dx = \frac{x^2}{2} \ln x – \int \! \frac{x^2}{2} \frac{1}{x} \ dx \)
\( \int \! x \ln x \ dx = \frac{x^2}{2} \ln x – \frac{x^2}{4} + C \)
\( \int \! \sin^{-1} x \ dx \)
\( u = \sin^{-1} x \ \ \ \ \ \ du = \frac{1}{\sqrt{1 – x^2}} \ dx\)
\( dv = dx \ \ \ \ \ \ v = x \)
\( \int \! \sin^{-1} x \ dx = x \sin^{-1} x – \int \! x \frac{1}{ \sqrt{1 – x^2 }} \ dx\)
\( \int \! \frac{x}{ \sqrt{1 – x^2 }} \ dx\)
\( t = 1 – x^2 \ \ \ \ \ \ – \frac{1}{2} \ dt = x \ dx\)
\( – \int \! \frac{1}{ 2 \sqrt{ t }} \ dt = – \sqrt{ t } + C\)
\( \int \! \sin^{-1} x \ dx = x \sin^{-1} x – \int \! x \frac{1}{ \sqrt{1 – x^2 }} \ dx = x \sin^{-1} x + \sqrt{ 1 – x^2 } + C \)
\( \int \! x e^{ x } \ dx \)
\( u = x \ \ \ \ \ \ du = dx \)
\( dv = e^{x} \ dx \ \ \ \ \ \ v = e^{x}\)
\( \int \! x e^{ x } \ dx = x e^{x} – \int \! e^{x} \ dx = x e^{x} – e^{x} + C\)
\( \int \! e^{ \sqrt{x} } \ dx \)
\( t = \sqrt{x} \ \ \ \ \ \ dt = \frac{1}{2 \sqrt{ x }} \ dx\)
\( 2 t \ dt = dx \)
\( \int \! 2 t e^{ t } \ dt \)
\( u = 2 t \ \ \ \ \ \ du = 2 \ dt \)
\( dv = e^{t} \ dt \ \ \ \ \ \ v = e^{t}\)
\( \int \! 2 t e^{ t } \ dt = 2 t e^{t} – \int \! 2 e^{t} \ dt = 2 t e^{t} – 2 e^{t} + K\)
\( \int \! e^{ \sqrt{x} } \ dx = 2 \sqrt{ x } e^{ \sqrt{ x } } -2 e^{ \sqrt{ x } } + C\)
\( \int \! e^{ x } \sin x \ dx \)
\( u = \sin x \ \ \ \ \ \ dv = e^{ x } \ dx\)
\( du = \cos x \ dx \ \ \ \ \ \ v = e^{ x } \)
\( \int \! e^{ x } \sin x \ dx = e^{ x } \sin x – \int \! e^{ x } \cos x \ dx \)
\( s = \cos x \ \ \ \ \ \ ds = – \sin x \ dx \)
\( dt = e^{x} \ dx \ \ \ \ \ \ t = e^{x}\)
\( \int \! e^{ x } \sin x \ dx = e^{ x } \sin x – \left( e^{ x } \cos x – \int \! – e^{ x } \sin x \ dx \right)\)
\( \int \! e^{ x } \sin x \ dx = e^{ x } \sin x – e^{ x } \cos x – \int \! e^{ x } \sin x \ dx \)
\( \int \! e^{ x } \sin x \ dx = \frac{1}{2} \left( e^{ x } \sin x – e^{ x } \cos x \right) + C \)