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Chapter 7: Problem 7

Evaluate the following integrals. $$\int \frac{d x}{(3-5 x)^{4}}$$

Short Answer

Expert verified
Question: Evaluate the integral of the given function: $$\int \frac{d x}{(3-5 x)^{4}}$$ Answer: The integral can be evaluated by following the steps of substitution, simplification, and integration. The final result is: $$\int \frac{d x}{(3-5 x)^{4}} = \frac{1}{15}(3-5x)^{-3} + C$$

Step by step solution

01

Identify an appropriate substitution

We can observe that the expression inside the parenthesis is linear in nature. We will make a substitution \(u = 3 - 5x\) to simplify the integrand.
02

Compute dx in terms of substitution's variable

Differentiating the substitution \(u = 3 - 5x\) with respect to \(x\), we get: $$\frac{d u}{d x} = -5$$ Now, let's solve for \(d x\) in terms of \(d u\): $$d x = \frac{d u}{-5}$$
03

Rewrite the integral with the substitution

Replacing \(dx\) and the expression \(3 - 5x\) with \(u\) in the integral, we get: $$\int \frac{1}{u^{4}} \left(\frac{d u}{-5}\right)$$ We can take out the constant -1/5 from the integral: $$-\frac{1}{5} \int \frac{d u}{u^{4}}$$
04

Evaluate the new integral

Now, we can integrate with respect to \(u\): $$ -\frac{1}{5} \int \frac{d u}{u^{4}} = -\frac{1}{5} \int u^{-4} d u$$ Using integration rules, we know that \(\int u^n du = \frac{u^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. In our case: $$ -\frac{1}{5} \int u^{-4} d u = -\frac{1}{5} \cdot \frac{u^{-3}}{-3} + C$$
05

Reverse the substitution

Now we need to reverse the substitution and replace \(u\) with \(3-5x\): $$-\frac{1}{5} \cdot \frac{(3-5x)^{-3}}{-3} + C = \frac{1}{15}(3-5x)^{-3} + C$$ Therefore, the solution for the given integral is: $$\int \frac{d x}{(3-5 x)^{4}} = \frac{1}{15}(3-5x)^{-3} + C$$

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