Problem 49 Use a change of variables to eva... [FREE SOLUTION]

Chapter 5: Problem 49

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{4} \frac{x}{x^{2}+1} d x$$

Short Answer

Expert verified

Explain the steps to solve the integral 鈭個鈦� (x/(x虏+1)) dx. 1. Perform a substitution by defining a new variable u = x虏 + 1. 2. Differentiate both sides of the equation with respect to x to find the relationship between dx and du (du = 2x dx; dx = (1/2x) du). 3. Adjust the limits of integration by plugging in the original x limits into the substitution equation (x = 0, u = 1; x = 4, u = 17). 4. Substitute u into the original integral and replace the relevant bounds and dx ((1/2) 鈭倎鹿鈦� (1/u) du). 5. Evaluate the definite integral, which leads to the final answer: (1/2) ln(17).

Step by step solution

Define the substitution

Define a new variable $u = x^2 + 1$. Now, we will differentiate both sides of this equation with respect to $x$ to find the relationship between $d x$ and $d u$.

Differentiate with respect to x

Differentiating both sides of the equation $u = x^2 + 1$ with respect to $x$, we get: $${d u} = {2xd x}\newline$$ Now, we can solve for ${d x}$: $$d x = \frac{1}{2x}d u$$

Adjust the limits of integration

Since our original limits of integration were in terms of $x$, we now need to find the corresponding limits for the new variable $u$. In the given integral, the bounds for $x$ are 0 and 4: At $x = 0$: $$u = (0)^2 + 1 = 1\newline$$ At $x = 4$: $$u = (4)^2 + 1 = 17$$ So the new limits of integration are $u = 1$ to $u = 17$.

Substitute and rewrite the integral

Now, we can substitute $u$ into the original integral and replace the bounds, as well as substitute $\frac{1}{2x}du$ for $dx$: $$\int_{0}^{4} \frac{x}{x^2+1}dx = \int_{1}^{17} \frac{1}{2u}du$$

Evaluate the integral

Now we can integrate with respect to $u$: $$\int_{1}^{17} \frac{1}{2u}du = \frac{1}{2} \left[ \ln|u| \right]_{1}^{17}$$ And find the value of this definite integral: $$\frac{1}{2} \left[ \ln|17| - \ln|1| \right] = \frac{1}{2} \ln \left|\frac{17}{1}\right| = \frac{1}{2} \ln(17)$$ So, the value of the integral $\int_{0}^{4} \frac{x}{x^2+1} dx$ is $\frac{1}{2} \ln(17)$.

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91影视

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{4} \frac{x}{x^{2}+1} d x$$

Short Answer

Step by step solution

Define the substitution

Differentiate with respect to x

Adjust the limits of integration

Substitute and rewrite the integral

Evaluate the integral

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