Find $\int \frac{4 x+1}{4 x^{2}+4 x} d x$ .
A) $\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C$
B) $\frac{1}{4} \ln |x|+\frac{3}{4} \ln |x+1|+C$
C) $\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C$.
D) $\ln |x|+\frac{1}{4} \ln |x+1|+C$

Question

Quizplus · Accepted Answer

The integral $\int \frac{4 x+1}{4 x^{2}+4 x} d x$ simplifies by factoring the denominator and splitting the fraction into simpler parts. Factoring the denominator gives $4x^2 + 4x = 4x(x + 1)$. The fraction can be decomposed as $\frac{A}{x} + \frac{B}{x+1}$, where $A$ and $B$ are constants to be determined. Solving for $A$ and $B$ gives $A = 1/4$ and $B = 3/4$, leading to the integral $\frac{1}{4} \ln |x|+\frac{3}{4} \ln |x+1|+C$.

Find ∫4x+14x2+4xdx\int \frac{4 x+1}{4 x^{2}+4 x} d x∫4x2+4x4x+1​dx A) 34ln⁡∣x∣+14ln⁡∣x+1∣+C\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C43​ln∣x∣+41​ln∣x+1∣+C

Find $\int \frac{4 x+1}{4 x^{2}+4 x} d x$
A) $\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C$