The integral $\int \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 5 x } d x$ is

A) $- \frac { 10 x + 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 }$
B) $\frac { 10 x - 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 } + C$
C) $\frac { 10 x + 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 } + C$
D) $\frac { 10 x - 29 \ln | 2 x - 5 | + 4 \ln | x | } { 20 } + C$
E) $\frac { 10 x + 29 \ln | 2 x - 5 | + 4 \ln | x | } { 20 } + C$

Question

The integral $\int \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 5 x } d x$ is

A) $- \frac { 10 x + 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 }$
B) $\frac { 10 x - 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 } + C$
C) $\frac { 10 x + 29 \ln | 2 x - 5 | - 4 \ln | x | } { 20 } + C$
D) $\frac { 10 x - 29 \ln | 2 x - 5 | + 4 \ln | x | } { 20 } + C$
E) $\frac { 10 x + 29 \ln | 2 x - 5 | + 4 \ln | x | } { 20 } + C$

Quizplus · Accepted Answer

To solve the integral $\int \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 5 x } d x$, we first perform polynomial long division or partial fraction decomposition if necessary. However, in this case, the numerator and the denominator are already in a suitable form for direct integration after simplifying. The integral can be approached by dividing the numerator by the denominator to simplify the expression, followed by integration. The correct answer involves recognizing the derivative of the logarithmic functions in the result, which corresponds to option C, indicating the integration process involves logarithmic functions with coefficients adjusted to match the derivative of the denominator's factors.

The Integral ∫x2+12x2−5xdx\int \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 5 x } d x∫2x2−5xx2+1​dx

The Integral $\int \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 5 x } d x$