Exercício - Integral - 2023-1

a) \int x \left(2x^2 + 3\right)^{10} \, dx b) \int \frac{x}{\left(x^2 + 5\right)^3} \, dx c) \int \frac{x}{e^x} \, dx d) \int \frac{1}{1+e^{3x}} \, dx e) \int \left(2x^3 + 3\right)e^{x^2} \, dx (a) \int x \cdot (2x^2+3)^{10} \, dx u = 2x^2 + 3 du/dx = 4x dx = du/4x \int x \cdot u^{10} \frac{du}{4x} \int \frac{u^{11}}{4 \cdot 11} = \frac{u^{11}}{44} = \frac{(2x^2+3)^{11}}{44} + C (b) \int \frac{x}{(x^2+5)^3} \, dx u = x^2 + 5 du/dx = 2x dx = du/2x \int \frac{x}{u^3} \frac{du}{2x} \int \frac{du}{2u^3} = \frac{1}{2} \int u^{-3} du = \frac{1}{2} \cdot \frac{u^{-2}}{-2} = \frac{-1}{4u^2} = \frac{-1}{4(x^2+5)^2} + C (c) \int \frac{x}{e^x} \, dx u = x dv = e^{-x} \, dx du = 1 \, dx v = -e^{-x} \int u \, dv = uv - \int v \, du = -x \cdot e^{-x} + \int e^{-x} \, dx = -x \cdot e^{-x} - e^{-x} + C = \frac{-(x+1)}{e^x} + C (d) \int \frac{dx}{1+e^{3x}} u = 3x du/dx = 3 dx = du/3 \int \frac{du}{3 \cdot (1+e^u)} w = e^u dw/du = e^u du = dw/e^u \frac{1}{3} \int \frac{dw}{w \cdot (w+1)} 1/3 \cdot \int \frac{1}{w} \cdot \frac{1}{w+1} \, dw = \frac{1}{3} \int \frac{dw}{w} - \frac{1}{3} \int \frac{dw}{w+1} w = T dw = dT \frac{1}{3} \int \frac{dT}{T} = \frac{1}{3} \ln|T| z = w+1 dz = dw -\frac{1}{3} \int \frac{dz}{z} = -\frac{1}{3} \ln|z| \frac{1}{3} \ln|T| - \frac{1}{3} \ln|z| + C = \frac{1}{3} \ln|e^{3x}| - \frac{1}{3} \cdot e^{3x+1}| = z = w+1 w = T w = e^u u = 3x \frac{1}{3} \cdot \ln\left|\frac{e^{3x}}{(e^{3x}+1)}\right| = = \frac{1}{3} \cdot 2\ln\left|\frac{e^{3x}}{e^{3x}+1}\right| + C = \frac{1}{3} \ln|e^{3x}| - \frac{1}{3} \ln(e^{3x+1})| = \frac{3x}{3} - \frac{\ln|e^{2x+1}|}{3} + C = \left(x - \frac{\ln|e^{3x+1}|}{3}\right) + C \oint (2x^2+1)e^{x^2} \, dx = = \oint 2e^{x^2} \cdot x^2 + e^{x^2} \, dx u = e^{x^2} du = 2e^{x^2} \cdot x dv = dx v = X = e^{x^2}\cdot x - \oint 2e^{x^2}\cdot x^2 + \oint 2e^{x^2} \cdot x^2 \, dx = = \boxed{e^{x^2} \cdot x + C}