Problem 5
Question
Find the integral. $$ \int \frac{x^{3}}{x^{2}+1} d x $$
Step-by-Step Solution
Verified Answer
The solution to the integral is \(0.5(x^{2}+1) + C\).
1Step 1: 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 \(\int \frac{0.5*u*du}{u}\).
2Step 2: Simplify the Integral and Evaluate
The integral simplifies to \(\int 0.5 du\) to which the answer is \(0.5u + C\).
3Step 3: Substitute Back
Change \(u\) back to the original variable. The final result is \(0.5(x^{2}+1) + C\).
Other exercises in this chapter
Problem 4
Find the general solution of the differential equation and check the result by differentiation. $$ \frac{d r}{d \theta}=\pi $$
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Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\cosh ^{-1} 2\) (b) \(\operatorname{sech}^{-1} \f
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In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{-1}^{1} x^{3} d x $$
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In Exercises \(5-18,\) evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{1} 2 x d x $$
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