(Use \(C\) for the constant of integration when applicable)
\[ \int _0^2 x \sin (3x) \d x \begin{prompt}=\answer {\frac {1}{9} \sin (6) - \frac {2}{3} \cos (6)}\end{prompt} \]
Make a substitution! Don’t forget to modify the limits of integration appropriately!
\[ \int _0^1 2x \sin (x) \d x \begin{prompt}=\answer {2 \sin (1) - 2 \cos (1)}\end{prompt} \]
\[ \int _0^3 (x+1) \sin \bigg (\frac {x}{2}\bigg ) \d x \begin{prompt}=\answer {2+4 \sin \left (\frac {3}{2}\right )-8 \cos \left (\frac {3}{2}\right )}\end{prompt} \]
\[ \ddx \int _0^{2x} 2t \sin (t) \d t \begin{prompt}=\answer {8x \sin (2x)}\end{prompt} \]
You can either explicitly calculate this or use the Fundamental Theorem of
Calculus!