Question

Make a substitution to express the integrand as a rational function and then evaluate the integral. (Use C for the constant of integration.)
∫ 2e^2x/ (e^2x +16e^x +63). dx

Answer 1

Answer:

$I = - 7 \cdot \ln|e^{x}+7| + 9 \cdot \ln|e^{x}+9| + C$

Step-by-step explanation:

Let consider the following algebraical substitution:

$u = e^{x}$

$du = e^{x} dx$

The integral is re-arranged to this form:

$I = 2 \cdot \int\ {\frac{u\ du}{u^{2}+16\cdot u + 63} } \, dx$

After doing some algebraical manipulation, the rational form of this integral is:

$I = -\int {\frac{7}{u + 7} } \, dx + \int {\frac{9}{u + 9} } \, dx$

The solution of this integral is:

$I = - 7 \cdot \ln|u+7| + 9 \cdot \ln|u+9| + C$

$I = - 7 \cdot \ln|e^{x}+7| + 9 \cdot \ln|e^{x}+9| + C$