I(a)=∫∞0sin2axx2⋅e4xdx diyelim.
I(a)=∫∞0sin2axx2⋅e4xdx⇒dIda=I′(a)=∫∞02x⋅sinax⋅cosaxx2⋅e4xdx⇒I′(a)=∫∞02⋅sinax⋅cosaxx⋅e4xdx⇒I′(a)=∫∞0sin2axx⋅e4xdx⇒dda(dIda)=I″(a)=∫∞02xcos2axx⋅e4xdx⇒I″(a)=2∫∞0cos2axe4xdx⇒I″(a)=2a2+4⇒I′(a)−I′(0)=∫a02x2+4dx⇒I′(a)=arctan(a2)⇒I(2)−I(0)=∫20arctan(x2)dx⇒I(2)=[−ln(x2+4)+xarctan(x2)+ln4]20⇒I(2)=π2−ln2.