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Question Number 129038 by bramlexs22 last updated on 12/Jan/21

$$\int \frac{{5\mathrm{e}}^{4\mathrm{t}}+{10\mathrm{e}}^{2\mathrm{t}}+2}{{\mathrm{e}}^{2\mathrm{t}}+2}\mathrm{dt}$$

Answered by liberty last updated on 12/Jan/21

$$\int \frac{({5\mathrm{e}}^{2\mathrm{t}}+1)({\mathrm{e}}^{2\mathrm{t}}+2)-{\mathrm{e}}^{2\mathrm{t}}}{{\mathrm{e}}^{2\mathrm{t}}+2}\mathrm{dt}=$$$$\int ({5\mathrm{e}}^{2\mathrm{t}}+1)\mathrm{dt}-\int \frac{{\mathrm{e}}^{2\mathrm{t}}}{{\mathrm{e}}^{2\mathrm{t}}+1}\mathrm{dt}=$$$$\frac{5}{2}{\mathrm{e}}^{2\mathrm{t}}+\mathrm{t}-\frac{1}{2}\int \frac{\mathrm{d}({\mathrm{e}}^{2\mathrm{t}}+2)}{{\mathrm{e}}^{2\mathrm{t}}+2}=$$$$\frac{5}{2}{\mathrm{e}}^{2\mathrm{t}}+\mathrm{t}-\frac{1}{2}\ln \mid {\mathrm{e}}^{2\mathrm{t}}+2\mid +\mathrm{C}$$

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