Answer:
[tex]I = - 7 \cdot \ln|e^{x}+7| + 9 \cdot \ln|e^{x}+9| + C[/tex]
Step-by-step explanation:
Let consider the following algebraical substitution:
[tex]u = e^{x}[/tex]
[tex]du = e^{x} dx[/tex]
The integral is re-arranged to this form:
[tex]I = 2 \cdot \int\ {\frac{u\ du}{u^{2}+16\cdot u + 63} } \, dx[/tex]
After doing some algebraical manipulation, the rational form of this integral is:
[tex]I = -\int {\frac{7}{u + 7} } \, dx + \int {\frac{9}{u + 9} } \, dx \\[/tex]
The solution of this integral is:
[tex]I = - 7 \cdot \ln|u+7| + 9 \cdot \ln|u+9| + C[/tex]
[tex]I = - 7 \cdot \ln|e^{x}+7| + 9 \cdot \ln|e^{x}+9| + C[/tex]
