Integration by Parts

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int t\sin t\ dt\ ;\ \ \ u=t,\ dv=\sin t\ dt$  

•  $t\cos^{-1}t-\left(1-t^2\right)^{\frac{1}{2}}+C$  

•  $-t\cos t+\sin t+C$  

•  $t\sin^{-1}t+\left(1-t^2\right)^{\frac{1}{2}}+C$  

•  $t\sin t+\cos t+C$  

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int xe^{4x}dx\ ;\ \ \ u=x,\ dv=e^{4x}dx$  

•  $-\frac{x}{4^x\ln4}-\frac{1}{4^x\cdot\left(\ln4\right)^2}+C$  

•  $\frac{x^5\ln x}{5}-\frac{x^5}{25}+C$  

•  $x\ln\left(x+4\right)-x+4\ln\left(x+4\right)+C$  

•  $\frac{xe^{4x}}{4}-\frac{e^{4x}}{16}+C$  

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int t\cdot2^t\ dt\ ;\ \ \ u=t,\ dv=2^tdt$  

•  $\frac{t^2\ln2t^2-t^2}{2}+C$  

•  $\frac{t\cdot2^t}{\ln2}-\frac{2^t}{\left(\ln2\right)^2}+C$  

•  $\frac{t^3\ln t}{3}-\frac{t^3}{9}+C$  

•  $\frac{2t^{\frac{3}{2}}\ln2t}{3}-\frac{4t^{\frac{3}{2}}}{9}+C$  

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int\log_5t\ dt\ ;\ \ \ u=\log_5t,\ dv=dt$  

•  $\frac{2t^2\ln3t-t^2}{4}+C$  

•  $-\frac{t}{3^t\ln3}-\frac{1}{3^t\cdot\left(\ln3\right)^2}+C$  

•  $t\ln\left(t^2+9\right)-2t+6\tan^{-1}\left(\frac{t}{3}\right)+C$  

•  $t\log_5t-\frac{t}{\ln5}+C$  

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int x^{4\ }\ln x\ dx\ ;\ \ u=\ln x,\ dv=x^4dx$  

•  $\frac{e^x}{4x+4}+C$  

•  $\frac{2x^{\frac{3}{2}}\ln4x}{3}-\frac{4x^{\frac{3}{2}}}{9}+C$  

•  $\frac{\left(4x^2-1\right)\cdot e^{4x^2}}{32}+C$  

•  $\frac{x^5\ln x}{5}-\frac{x^5}{25}+C$  

Evaluate the indefinite integral using integration by parts. u and dv are provided.
 $\int x\cos2x\ dx\ ;\ \ u=x,\ dv=\cos2x\ dx$  

•  $-\frac{x\cos2x}{2}+\frac{\sin2x}{4}+C$  

•  $\frac{x\sin2x}{2}+\frac{\cos2x}{4}+C$  

•  $x\sin^{-1}2x+\frac{\left(1-4x^2\right)^{\frac{1}{2}}}{2}+C$  

•  $-\frac{x\cot2x}{2}+\frac{\ln\sin2x}{4}+C$  

Evaluate the indefinite integral using integration by parts. 
 $\int3x\ e^{2x}\ dx\$  

•  $-\frac{xe^{2x}}{2}+\frac{e^{2x}}{4}+C$  

•  $\frac{3xe^{2x}}{2}-\frac{3e^{2x}}{4}+C$  

•  $xe^{-2x}+\frac{\left(1-x^2\right)^{ }}{2}+C$  

•  $-\frac{xe^{2x}}{2}+\frac{\ln e^{2x}}{4}+C$  

Evaluate the indefinite integral using integration by parts. 
 $\int t^2\ln t\ dt\$  

•  $\frac{2t^2\ln2t-t^2}{4}+C$  

•  $\frac{t^3\ \ln3}{3}-\frac{t^3}{9}+C$  

•  $\frac{e^t}{2t+2}+C$  

•  $\frac{-2t-1}{4e^{2t}}+C$