Evaluate the integral by using a substitution prior to integration by parts.
-$$\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x$$
A) $\ln \left( 10 x + 10 x ^ { 2 } \right) - 2 x + C$
B) $\ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) + C$
C) $x \ln \left( 10 x + 10 x ^ { 2 } \right) + C$
D) $x \ln \left( 10 x + 10 x ^ { 2 } \right) + \ln ( x + 1 ) - 2 x + C$

Question

Quizplus · Accepted Answer

First, factor out the common term inside the logarithm: $\ln(10x + 10x^2) = \ln(10x(1 + x))$. Then, use substitution $u = 10x + 10x^2$, so $du = (10 + 20x)dx$, but this doesn't directly lead to a simple substitution. Instead, integrate by parts with $u = \ln(10x + 10x^2)$ and $dv = dx$, leading to $du = \frac{10 + 20x}{10x + 10x^2}dx$ and $v = x$. This process, along with recognizing the derivative of $x + 1$ within the integral, leads to the correct answer, which includes both the original logarithmic function multiplied by $x$ and adjustments for the integration process, including $\ln(x + 1)$ and a linear term in $x$.

Evaluate the Integral by Using a Substitution Prior to Integration ∫ln⁡(10x+10x2)dx\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x∫ln(10x+10x2)dx

Evaluate the Integral by Using a Substitution Prior to Integration $\int \ln \left( 10 x + 10 x ^ { 2 } \right) d x$