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Question Number 83927 by M±th+et£s last updated on 08/Mar/20

$$\int \frac{sinh\left(x\right)+e^{3x}}{sinh\left(x\right)-e^x}dx$$

Answered by TANMAY PANACEA last updated on 08/Mar/20

$$\int \frac{\frac{e^x-e^{-x}}{2}+e^{3x}}{\frac{e^x-e^{-x}}{2}-e^x}dx$$$$=\int \frac{e^x-e^{-x}+2e^{3x}}{e^x-e^{-x}-2e^x}dx$$$$=\int \frac{e^{2x}-1+2e^{4x}}{-e^{2x}-1}dx$$$$=\int \frac{1-2e^{4x}-e^{2x}}{e^{2x}+1}dx$$$$t=e^{2x}+1\to dt=e^{2x}\times 2\times dx$$$$\frac{dt}{2(t-1)}=dx$$$$\int \frac{1-2{(t-1)}^2-(t-1)}{t}\times \frac{dt}{2(t-1)}$$$$\frac{1}{2}\int \frac{1-2{(t-1)}^2-(t-1)}{t(t-1)}dt$$$$\frac{1}{2}\int \frac{t-(t-1)}{t(t-1)}dt-\int \frac{t-1}{t}dt-\frac{1}{2}\int \frac{dt}{t}$$$$\frac{1}{2}\int \frac{dt}{t-1}-\frac{1}{2}\int \frac{dt}{t}-\int dt+\int \frac{dt}{t}-\frac{1}{2}\int \frac{dt}{t}$$$$\frac{1}{2}ln(t-1)-t+c$$$$\frac{1}{2}ln\left(e^{2x}\right)-(e^{2x}+1)+c$$$$x-(e^{2x}+1)+c$$

Commented by M±th+et£s last updated on 08/Mar/20

$$thankyousirnicesolution$$

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