Find the definite integral. $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x$
A) $\frac { 2048 \ln 4 - 255 } { 16 }$
B) $\frac { 1024 \ln 4 - 255 } { 16 }$
C) $\frac { - 1024 \ln 4 - 255 } { 16 }$
D) $\frac { 1024 \ln 4 + 255 } { 16 }$
E)none of the above

Question

Quizplus · Accepted Answer

To solve the integral $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x$, we can use integration by parts, where we let $u = \ln x$ and $dv = x^3 dx$. This gives us $du = \frac{1}{x} dx$ and $v = \frac{x^4}{4}$. Applying the integration by parts formula $\int u dv = uv - \int v du$, we get:$$\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x = \left. \frac{x^4}{4} \ln x \right|_1^4 - \int_1^4 \frac{x^4}{4} \cdot \frac{1}{x} dx = \left. \frac{x^4}{4} \ln x \right|_1^4 - \frac{1}{4} \int_1^4 x^3 dx$$Evaluating the integral and the limits, we find:$$\frac{256}{4} \ln 4 - \left( \frac{1}{4} \cdot \frac{x^4}{4} \right) \Big|_1^4 = 64 \ln 4 - \frac{256}{16} + \frac{1}{16} = 64 \ln 4 - 16 + \frac{1}{16}$$$$= 64 \ln 4 - \frac{255}{16} = \frac{1024 \ln 4 - 255}{16}$$Therefore, the correct answer is B) $\frac { 1024 \ln 4 - 255 } { 16 }$.

Find the Definite Integral ∫14x3ln⁡xdx\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x∫14​x3lnxdx

Find the Definite Integral $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x$