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Question Number 143210 by ArielVyny last updated on 11/Jun/21

$${\int }_0^1\frac{7^{x+1}+3^{x+1}}{x+1}dx$$

Answered by MJS_new last updated on 11/Jun/21

$$\int \frac{7^{x+1}++3^{x+1}}{x+1}dx=\int \frac{7^{x+1}}{x+1}dx+\int \frac{3^{x+1}}{x+1}dx$$$$$$$$\int \frac{b^{x+1}}{x+1}dx=$$$$[t=(x+1)\ln b\to dx=\frac{dt}{\ln b}]$$$$=\int \frac{{\mathrm{e}}^t}{t}dt=\mathrm{Ei}t=\mathrm{Ei}\left((x+1)\ln b\right)+C$$$$\Rightarrow \mathrm{answer}\mathrm{is}\approx 17.0130$$

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