int frac{30x^{14} + 15^x log 225}{x^{15} + 15 | vedclass

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Question

(C) Let $I = \int \frac{30x^{14} + 15^x \log 225}{x^{15} + 15^x} dx$.Note that $15^x \log 225 = 15^x \log(15^2) = 2 \cdot 15^x \log 15$.Let $u = x^{15

Vedclass · Accepted Answer

$\log(x^{15} + 15^x)^2 + c$, where $c$ is the constant of integration

Vedclass · Answer

(C) Let $I = \int \frac{30x^{14} + 15^x \log 225}{x^{15} + 15^x} dx$.Note that $15^x \log 225 = 15^x \log(15^2) = 2 \cdot 15^x \log 15$.Let $u = x^{15} + 15^x$.Then $du = (15x^{14} + 15^x \log 15) dx$.Observe that the numerator is $2(15x^{14} + 15^x \log 15) dx = 2 du$.Thus, $I = \int \frac{2}{u} du = 2 \log|u| + c$.Substituting $u$ back, $I = 2 \log|x^{15} + 15^x| + c = \log(x^{15} + 15^x)^2 + c$.

$\int \frac{30x^{14} + 15^x \log 225}{x^{15} + 15^x} dx =$

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