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Consider the indefinite integral
6 + e^x
The most appropriate substitution to simplify this integral is u = f(x) where f(x) = e^x+6
We then have
dx = g(u) du
where
g(u) = e^-u
Hint: you need to back substitute for x in terms of u for this part.
After substituting into the original integral we obtain
h(u) du where
h(u) = 1/(1-6/u)^2
To evaluate this integral, rewrite the numerator as
6 = u – (u – 6)
simplify; then integrate, thus obtaining
h(u) du = H(u)
where H(u) = 1/6 * ln(1-6/u)
After substituting back for u, we obtain our final answer: ∫(e^x+6) dx = -ln(e^x+6) + 6 + C
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Transcript
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