QuestionExercise 10.3
1. Use the integration by parts to evaluate the following integrals. (a) $\int s e^{-2 s} d s$ (b) $\int \ln (x+1) d x$ (c) $\int t \sin 2 t d t$ (d) $\int x 2^{x} d x$ (e) $\int x \cos 5 x d x$ (f) $\int e^{x} \cos x d x$

Studdy Solution
Rearrange to solve for $\int e^x \cos x \, dx$: $$2 \int e^x \cos x \, dx = e^x (\cos x + \sin x)$$ $$\int e^x \cos x \, dx = \frac{1}{2} e^x (\cos x + \sin x) + C$$
The solutions to the integrals are: (a) $-\frac{1}{2} s e^{-2s} - \frac{1}{4} e^{-2s} + C$ (b) $x \ln(x+1) - (x+1) + \ln(x+1) + C$ (c) $-\frac{1}{2} t \cos 2t + \frac{1}{4} \sin 2t + C$ (d) $\frac{x 2^x}{\ln 2} - \frac{1}{(\ln 2)^2} 2^x + C$ (e) $\frac{x}{5} \sin 5x + \frac{1}{25} \cos 5x + C$ (f) $\frac{1}{2} e^x (\cos x + \sin x) + C$