Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$
A) $2 \sqrt { t ^ { 3 } + 3 } + C$
B) $\sqrt { t ^ { 3 } + 3 } + C$
C) $2 \sqrt { t ^ { 2 } + 3 } + C$
D) $\frac { 1 } { 2 } \sqrt { t ^ { 3 } + 3 } + C$
E) $\frac { 1 } { 2 } \sqrt { t ^ { 2 } + 3 } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$, let $u = t^3 + 3$. Then, $du = 3t^2 dt$. Substituting $u$ and $du$ into the integral gives $\int \frac{1}{\sqrt{u}} du$, which integrates to $2\sqrt{u} + C = 2\sqrt{t^3 + 3} + C$. Differentiating $2\sqrt{t^3 + 3} + C$ with respect to $t$ returns the original function, confirming the correctness of the solution.

Find the Indefinite Integral of the Following Function and Check ∫3t2t3+3dt\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t∫t3+3​3t2​dt

Find the Indefinite Integral of the Following Function and Check $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$