After applying integration by parts to ∫ x^3 ln(x) dx, which term represents uv?

After applying integration by parts to ∫ x^3 ln(x) dx, which term represents uv?

• A. (x^4/4) ln(x)
• B. (x^5/5) ln(x)
• C. (x^3/3) ln(x)
• D. (x^4) ln(x)

Answer: (A)

To identify which term represents \( uv \) after applying integration by parts to the integral \( \int x^3 \ln(x) \, dx \), we first need to understand how the integration by parts formula works. The formula is given by: \[ \int u \, dv = uv - \int v \, du \] In this context, we want to select \( u \) and \( dv \) wisely. A common choice when dealing with \( \ln(x) \) in integrals is to let: - \( u = \ln(x) \) (which differentiates to \( du = \frac{1}{x} \, dx \)), - \( dv = x^3 \, dx \) (which integrates to \( v = \frac{x^4}{4} \)). Once we apply the integration by parts, we substitute \( u \) and \( v \) back into the formula. The term \( uv \) would be calculated as follows: \[ uv = \ln(x) \cdot \frac{x^4}{4} \] So, \( uv \) is exactly \( \frac{x^4}{4} \ln(x) \).

To identify which term represents ( uv ) after applying integration by parts to the integral ( \int x^3 \ln(x) , dx ), we first need to understand how the integration by parts formula works. The formula is given by:

[

\int u , dv = uv - \int v , du

]

In this context, we want to select ( u ) and ( dv ) wisely. A common choice when dealing with ( \ln(x) ) in integrals is to let:

• ( u = \ln(x) ) (which differentiates to ( du = \frac{1}{x} , dx )),

• ( dv = x^3 , dx ) (which integrates to ( v = \frac{x^4}{4} )).

Once we apply the integration by parts, we substitute ( u ) and ( v ) back into the formula. The term ( uv ) would be calculated as follows:

[

uv = \ln(x) \cdot \frac{x^4}{4}

]

So, ( uv ) is exactly ( \frac{x^4}{4} \ln(x) ).