Find the indefinite integral of the following function and check the result by differentiation. $\int x ^ { 4 } \sqrt { \left( 4 + x ^ { 5 } \right) } d x$
A) $\frac { \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 10 } + C$
B) $\frac { 2 \left( 4 + x ^ { 5 } \right) ^ { \frac { 2 } { 3 } } } { 15 } + C$
C) $\frac { \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 15 } + C$
D) $\frac { 2 \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 15 } + C$
E)none of the above

Question

Quizplus · Accepted Answer

The integral can be solved using substitution: let $ u = 4 + x^5 $, then $ du = 5x^4 dx $. The integral becomes $\int \frac{\sqrt{u}}{5} du$, which evaluates to $\frac{2}{15} u^{3/2} + C$. Substituting back $ u = 4 + x^5 $, we get $\frac{2}{15} (4 + x^5)^{3/2} + C$.

Find the Indefinite Integral of the Following Function and Check ∫x4(4+x5)dx\int x ^ { 4 } \sqrt { \left( 4 + x ^ { 5 } \right) } d x∫x4(4+x5)​dx

Find the Indefinite Integral of the Following Function and Check $\int x ^ { 4 } \sqrt { \left( 4 + x ^ { 5 } \right) } d x$