Integration by Parts — Sean's Space

Integration by Parts - practice

$\int x \cos (π x) d x; u = x, d v = \cos (π x) d x$

$\frac{1}{π} x \sin (π x) + \frac{1}{π^{2}} \cos (π x) + C$

$\int x e^{2 x} d x; u = x, d v = e^{2 x} d x$

$\frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + C$

$\int x \cos (π x) d x; u = x, d v = \cos (π x) d x$

$\frac{1}{π} x \sin (π x) + \frac{1}{π^{2}} \cos (π x) + C$

$\int \sin^{- 1} x d x; u = \sin^{- 1} x, d v = d x$

$x \arcsin (x) + \sqrt{1 - x^{2}} + C$

$\int \frac{\ln x}{x^{2}} d x$

$- \frac{\ln (x)}{x} - \frac{1}{x} + C$

$\int x \sin 10 x d x$

$- \frac{1}{10} x \cos (10 x) + \frac{1}{100} \sin (10 x) + C$

$\int y e^{- y} d y$

$- e^{- y} y - e^{- y} + C$

$\int \cos^{- 1} x d x$

$x \arccos (x) - \sqrt{1 - x^{2}} + C$

$\int_{0}^{1} x 3^{x} d x$

$\frac{3 \ln (3) - 2}{\ln^{2} (3)}$

$\int \frac{z}{10^{z}} d z$

$- \frac{10^{- z} z}{\ln (10)} - \frac{1}{\ln^{2} (10)} \cdot 10^{- z} + C$

$\int \tan^{- 1} (2 y) d y$

$- \frac{1}{2} \ln | \cos (2 y) | + C$

$\int (\ln x)^{2} d x$

$x \ln^{2} (x) - 2 (x \ln (x) - x) + C$

$\int \ln \sqrt{x} d x$

$\frac{1}{2} x \ln (x) - \frac{1}{2} x + C$

$\int t^{2} \sin β t d t$

$- \frac{1}{β} t^{2} \cos (β t) + \frac{2}{β^{3}} (β t \sin (β t) + \cos (β t)) + C$

$\int e^{3 x} \cos x d x$

$\frac{e^{3 x} \sin (x)}{10} + \frac{3 e^{3 x} \cos (x)}{10} + C$

$\int e^{x} \sin (π x) d x$

$- \frac{π e^{x} \cos (π x)}{π^{2} + 1} + \frac{e^{x} \sin (π x)}{π^{2} + 1} + C$

$\int_{0}^{π} x \sin x \cos x d x$

$- \frac{π}{4}$

$\int_{0}^{π / 3} \sin x \ln (\cos x) d x$

$- \frac{1}{2} + \frac{1}{2} \ln (2)$

$\int x \ln (1 + x) d x$

$\begin{aligned} \frac{1}{2} \ln (1 + x) (1 + x)^{2} - \ln (1 + x) - x \ln (1 + x) \\ - \frac{1}{4} (1 + x)^{2} + 1 + x + C \end{aligned}$

$\int_{\sqrt{π / 2}}^{\sqrt{π}} θ^{3} \cos (θ^{2}) d θ$

$\frac{- 2 - π}{4}$

$\int \frac{\arcsin (\ln x)}{x} d x$

$\ln (x) \arcsin (\ln (x)) + \sqrt{1 - \ln^{2} (x)} + C$

$\int \cos (\ln x) d x$

$\frac{1}{2} x \sin (\ln (x)) + \frac{1}{2} x \cos (\ln (x)) + C$