1. Evaluate the following integrals: (a) $int x sin x , dx$ (b) $int x cos 4x , dx$ (c) $int xe^{-x} , dx$ (d) $int x^2 cos x , dx$ (e) $int 2x^2 e^x , dx$ (f) $int x^2 ln |x| , dx$ (g) $int an^{-1} x , dx$ (h) $int sin^{-1} x , dx$ (i) $int e^x cos x , dx$ (j) $int sin^3 x , dx$ 2. Calculate the value of each of the following: (a) $int_0^{pi} x cos frac{1}{2} x , dx$ (b) $int_0^1 x^2 e^x , dx$ (c) $int_1^2 x^3 ln |x| , dx$ (d) $int_0^{pi/4} x^2 sin 2x , dx$ (e) $int_0^1 x an^{-1} x , dx$
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Step 1
Let $u = r$ and $dv = \sin{r} dr$. Then, $du = dr$ and $v = -\cos{r}$. Using the integration by parts formula, we get: $\int r\sin{r} dr = -r\cos{r} - \int -\cos{r} dr = -r\cos{r} + \sin{r} + C$ Show more…
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