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Chapter 17: Problem 13

Find \(\int \frac{2 x+7}{x^{2}+7 x} \mathrm{~d} x\).

Short Answer

Expert verified
Question: Evaluate the integral \(\int \frac{2x+7}{x^2+7x} dx\). Answer: \(\int \frac{2x+7}{x^2+7x} dx = \ln|x^2+7x| + C\), where \(C\) is an integration constant.

Step by step solution

01

Identify the substitution

Let's look at our integral, $$\int \frac{2x+7}{x^2+7x} dx.$$ Notice that if we differentiate the denominator -\(x^2 + 7x\)-, we obtain almost exactly the numerator. Specifically, $$\frac{d}{dx}(x^2+7x) = 2x + 7.$$ Thus, we will make the substitution \(u=x^2+7x\).
02

Compute du in terms of dx

Now, let's differentiate \(u\) with respect to \(x\) and isolate \(dx\). $$\frac{du}{dx} = 2x + 7.$$ So, $$du = (2x + 7)dx.$$
03

Rewrite the integral in terms of u

Now that we have the substitution and its differential, we can rewrite the integral in terms of \(u\): $$\int \frac{2x+7}{x^2+7x} dx = \int \frac{1}{u} du.$$
04

Evaluate the integral in terms of u

This new integral is straightforward to evaluate: $$\int \frac{1}{u} du = \ln|u|+C,$$ where \(C\) is an integration constant.
05

Re-substitute in terms of x

Finally, we substitute the original expression for \(u\) back into the integral: $$\ln|u|+C = \ln|x^2+7x|+C.$$
06

Simplify and write the final answer

Writing the answer more neatly, we get: $$\int \frac{2x+7}{x^2+7x} dx = \ln|x^2+7x| + C.$$

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Most popular questions from this chapter

Find \(\int 3 \cos n \pi x \mathrm{~d} x\).

Find the following integrals using an appropriate technique selected from this chapter: (a) \(\int \frac{1}{4+x^{2}} \mathrm{~d} x\) (b) \(\int \cos ^{2} x \mathrm{~d} x\) (c) \(\int \frac{1}{4-t} \mathrm{~d} t\) (d) \(\int_{0}^{\pi} \cos t \mathrm{~d} t\) (e) \(\int_{1}^{1.2} \tan x \mathrm{~d} x\) (f) \(\int \frac{2 x^{2}+x+2}{x^{3}+x} \mathrm{~d} x\) (g) \(\int \sin 3 t \cos 3 t \mathrm{~d} t\) (h) \(\int \frac{1}{16-x^{2}} \mathrm{~d} x\) (i) \(\int \frac{3}{\sqrt{t^{2}+0.25}} \mathrm{~d} t\) (j) \(\int_{-1}^{1} \cosh 3 t \mathrm{~d} t\) (k) \(\int_{1}^{3} \frac{1}{1+x^{2}} \mathrm{~d} x\) (1) \(\int \frac{1}{\sqrt{100-4 x^{2}}} \mathrm{~d} x\) (m) \(\int x \cos \frac{x}{2} \mathrm{~d} x\) (n) \(\int 3^{x} \mathrm{~d} x\) (o) \(\int \sec t \mathrm{~d} t\) (p) \(\int \sin 5 x \cos 2 x \mathrm{~d} x\) (q) \(\int \frac{1}{x(x-3)} \mathrm{d} x\) (r) \(\int \frac{1}{t^{2}(t-1)} \mathrm{d} t\) (s) \(\int \frac{\ln x}{x} \mathrm{~d} x\) (t) \(\int t \sqrt{3 t-2} \mathrm{~d} t\) (u) \(\int \frac{1}{5+4 \cos x} \mathrm{~d} x\) (v) \(\int_{0}^{2} t^{2} \mathrm{e}^{t} \mathrm{~d} t\) (w) \(\int \cos x \sqrt{\sin x} \mathrm{~d} x\) (x) \(\int \frac{x^{2}}{x^{3}+1} \mathrm{~d} x\) (y) \(\int \frac{8 t+10}{4 t^{2}+8 t+3} \mathrm{~d} t\) (z) \(\int \frac{7}{\left(\mathrm{e}^{t}\right)^{7}} \mathrm{~d} t\)

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$

Find \(\int_{-\infty}^{0} \mathrm{e}^{7 x} \mathrm{~d} x\)

Find \(\int \frac{\sin ^{3} x}{\cos x}+\sin x \cos x \mathrm{~d} x\).
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