Uppgift 4

(a) Med variabelsubstitutionen $e^x=t$, dvs. $x=\ln t$ och $dx=dt/t$, får vi $$\int(e^{2x}-e^{3x})\sin(e^x)\,dx=\int(t^2-t^3)\sin t\,\frac{dt}{t}=\int(t-t^2)\sin t\,dt.$$ Partiell integration ger $$=(t^2-t)\cos t+(1-2t)\sin t-2\cos t+C,$$ dvs. $$\bigl(e^{2x}-e^x-2\bigr)\cos(e^x)+(1-2e^x)\sin(e^x)+C.$$

(b) $$\int_1^4\frac{3x^2+\frac1{2\sqrt x}}{x^3+\sqrt x+2}\,dx=\left[\ln\left(x^3+\sqrt x+2\right)\right]_1^4=\ln(64+2+2)-\ln4=\ln17.$$