Integration by Parts - practice
\[ \int x \cos (\pi x) d x ; \quad u=x, d v=\cos (\pi x) d x\]
\[ \frac{1}{\pi} x \sin (\pi x)+\frac{1}{\pi^2} \cos (\pi x)+C\]
\[\int x e^{2 x} d x ; \quad u=x, d v=e^{2 x} d x\]
\[\frac{1}{4}\left(e^{2 x} \cdot 2 x-e^{2 x}\right)+C\]
\[ \int x \cos (\pi x) d x ; \quad u=x, d v=\cos (\pi x) d x\]
\[ \frac{1}{\pi} x \sin (\pi x)+\frac{1}{\pi^2} \cos (\pi x)+C\]
\[\int \sin ^{-1} x d x ; \quad u=\sin ^{-1} x, d v=d x\]
\[x \arcsin (x)+\sqrt{1-x^2}+C\]
\[\int \frac{\ln x}{x^2} d x\]
\[-\frac{\ln (x)}{x}-\frac{1}{x}+C\]
\[\int x \sin 10 x d x\]
\[-\frac{1}{10} x \cos (10 x)+\frac{1}{100} \sin (10 x)+C\]
\[\int y e^{-y} d y\]
\[-e^{-y} y-e^{-y}+C\]
\[\int \cos ^{-1} x d x\]
\[x \arccos (x)-\sqrt{1-x^2}+C\]
\[ \int_0^1 x 3^x d x \]
\[ \frac{3 \ln (3)-2}{\ln ^2(3)} \]
\[ \int \frac{z}{10^z} d z \]
\[ -\frac{10^{-z} z}{\ln (10)}-\frac{1}{\ln ^2(10)} \cdot 10^{-z}+C \]
\[ \int \tan ^{-1}(2 y) d y \]
\[ -\frac{1}{2} \ln |\cos (2 y)|+C \]
\[ \int(\ln x)^2 d x \]
\[ x \ln ^2(x)-2(x \ln (x)-x)+C \]
\[ \int \ln \sqrt{x} d x \]
\[ \frac{1}{2} x \ln (x)-\frac{1}{2} x+C \]
\[ \int t^2 \sin \beta t d t \]
\[ -\frac{1}{\beta} t^2 \cos (\beta t)+\frac{2}{\beta^3}(\beta t \sin (\beta t)+\cos (\beta t))+C \]
\[ \int e^{3 x} \cos x d x \]
\[ \frac{e^{3 x} \sin (x)}{10}+\frac{3 e^{3 x} \cos (x)}{10}+C \]
\[ \int e^x \sin (\pi x) d x \]
\[ -\frac{\pi e^x \cos (\pi x)}{\pi^2+1}+\frac{e^x \sin (\pi x)}{\pi^2+1}+C \]
\[ \int_0^\pi x \sin x \cos x d x \]
\[ -\frac{\pi}{4} \]
\[ \int_0^{\pi / 3} \sin x \ln (\cos x) d x \]
\[ -\frac{1}{2}+\frac{1}{2} \ln (2) \]
\[ \int x \ln (1+x) d x \]
\[ \small \begin{align*} &\frac{1}{2} \ln (1+x)(1+x)^2-\ln (1+x) -x \ln (1+x)\\ &\hspace{60mm} -\frac{1}{4}(1+x)^2+1+x+C \end{align*} \]
\[ \int_{\sqrt{\pi / 2}}^{\sqrt{\pi}} \theta^3 \cos \left(\theta^2\right) d \theta \]
\[ \frac{-2-\pi}{4} \]
\[ \int \frac{\arcsin (\ln x)}{x} d x \]
\[ \ln (x) \arcsin (\ln (x))+\sqrt{1-\ln ^2(x)}+C \]
\[ \int \cos (\ln x) d x \]
\[ \frac{1}{2} x \sin (\ln (x))+\frac{1}{2} x \cos (\ln (x))+C \]
\[ Hard Q1]
\[ Hard A1]
\[Hard Q2]
\[Hard A2]