Calculate $\int \frac{d t}{3+\sqrt{t}}$ .
A) $2(3+\sqrt{t})+6 \ln (3+\sqrt{t})+C$
B) $2(3+\sqrt{t})-6 \ln (3+\sqrt{t})+C$
C) $\frac{\sqrt{t} \ln (3+\sqrt{t})}{2}+C$
D) $-\frac{\sqrt{t} \ln (3+\sqrt{t})}{2}+C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \frac{d t}{3+\sqrt{t}}$, let $u = 3 + \sqrt{t}$, which implies $du = \frac{1}{2\sqrt{t}} dt$, or $2 du = \frac{dt}{\sqrt{t}}$. Substituting $u$ and rearranging for $dt$, we get the integral in terms of $u$, which simplifies to a basic form that can be integrated directly. The correct solution involves recognizing the structure of the integral and applying substitution correctly, leading to an answer that involves $u$ and natural logarithms, but the precise manipulation and integration yield $2(3+\sqrt{t})-6 \ln (3+\sqrt{t})+C$, which corresponds to option B.

Calculate ∫dt3+t\int \frac{d t}{3+\sqrt{t}}∫3+t​dt​ A) 2(3+t)+6ln⁡(3+t)+C2(3+\sqrt{t})+6 \ln (3+\sqrt{t})+C2(3+t​)+6ln(3+t​)+C B) 2(3+t)−6ln⁡(3+t)+C2(3+\sqrt{t})-6 \ln (3+\sqrt{t})+C2(3+t​)−6ln(3+t​)+C

Calculate $\int \frac{d t}{3+\sqrt{t}}$
A) $2(3+\sqrt{t})+6 \ln (3+\sqrt{t})+C$ B) $2(3+\sqrt{t})-6 \ln (3+\sqrt{t})+C$