/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 5

Find the integral. $$ \int \frac{x^{3}}{x^{2}+1} d x $$

Short Answer

Expert verified
The solution to the integral is \(0.5(x^{2}+1) + C\).

Step by step solution

01

Express the denominator function in terms of its derivative

Firstly, observe that the derivative of \(x^{2} + 1\) is \(2x\), which can be found in the numerator. So, if we let \(u = x^{2}+1\), then du = \(2x dx\). However, we need \(x dx\), not \(2x dx\). To adjust for this, we can write \(x dx = 0.5 * du\). The integral now becomes \(\int \frac{0.5*u*du}{u}\).
02

Simplify the Integral and Evaluate

The integral simplifies to \(\int 0.5 du\) to which the answer is \(0.5u + C\).
03

Substitute Back

Change \(u\) back to the original variable. The final result is \(0.5(x^{2}+1) + C\).

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