/*! 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 6 \(\int\left(x^{2}-2 x+3\right) \... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 6

\(\int\left(x^{2}-2 x+3\right) \ell n x d x\)

Short Answer

Expert verified
Question: Find the integral of the function (x^2 - 2x + 3)ln(x) with respect to x. Answer: The integral of the function (x^2 - 2x + 3)ln(x) with respect to x is given by the expression: $$\left(\dfrac{1}{3}x^3 - x^2 + 3x\right)\ell n x - \left(\dfrac{1}{9}x^3 - \dfrac{1}{2}x^2 +3x\right) + C$$ where C is the constant of integration.

Step by step solution

01

Choose u and dv

To choose our u and dv, we will use the following guideline: choose u as the function that simplifies when differentiated, and choose dv as the function that is easy to integrate. In this case, we choose: $$u = \ell n x \implies du = \dfrac{1}{x}dx$$ $$dv = (x^2 - 2x + 3)dx \implies v = \int(x^2 - 2x + 3)dx = \dfrac{1}{3}x^3 - x^2 + 3x$$
02

Apply Integration by Parts

Now, we will apply the integration by parts formula: $$\int u dv = uv - \int v du$$ Substitute the values we found in Step 1: $$\int (x^2 - 2x + 3)\ell n x dx = \left(\dfrac{1}{3}x^3 - x^2 + 3x\right)\ell n x - \int \left(\dfrac{1}{3}x^3-x^2+3x\right)\dfrac{1}{x} dx$$ Simplify the integral on the right by dividing each term by x: $$\left(\dfrac{1}{3}x^3 - x^2 + 3x\right)\ell n x - \int \left(\dfrac{1}{3}x^2 - x + 3\right) dx$$
03

Integrate the Remaining Polynomial

Now, we need to integrate the remaining polynomial: $$\int \left(\dfrac{1}{3}x^2 - x + 3\right) dx= \dfrac{1}{9}x^3 - \dfrac{1}{2}x^2 +3x$$
04

Write the Final Solution

Substitute the integral back into the equation from Step 2, and add the constant of integration C: $$\int (x^2 - 2x + 3)\ell n x dx = \left(\dfrac{1}{3}x^3 - x^2 + 3x\right)\ell n x - \left(\dfrac{1}{9}x^3 - \dfrac{1}{2}x^2 +3x\right) + C$$ This is the final solution for the given integral.

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Most popular questions from this chapter

Evaluate the following integrals : (i) \(\int \frac{(\sqrt{x}+1)\left(x^{2}-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}} d x\) (ii) \(\int \frac{\sqrt{1-\mathrm{x}^{2}}-1}{\mathrm{x}}\left(\frac{1-\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}+\mathrm{x}-1}+\frac{\sqrt{1+\mathrm{x}}}{\sqrt{1+\mathrm{x}}-\sqrt{1-\mathrm{x}}}\right) \mathrm{dx}\) (iii) \(\int \frac{x^{4}+5 x^{3}+15 x-9}{\frac{x^{6}+3 x^{4}}+\frac{9}{x^{4}}}{\left(x^{3}-4 x+3 x^{2}-12\right) / x^{5}} d x\) (iv) \(\int \frac{\sqrt[3]{x+\sqrt{2-x^{2}}} \sqrt[6]{1-x \sqrt{2-x^{2}}}}{\sqrt[3]{1-x^{2}}} d x\)

Evaluate the following integrals: (i) \(\int \frac{d x}{x^{3} \sqrt{1-x^{2}}}\) (ii) \(\int \frac{x^{4} d x}{\left(a^{2}+x^{2}\right)^{2}}\) (iii) \(\int \frac{x^{2} d x}{\left(a+c x^{2}\right)^{7 / 2}}\) (iv) \(\int \frac{x^{3} d x}{\left(a^{2}+x^{2}\right)^{3 / 2}}\)

Evaluate the following integrals: (i) \(\int \frac{\sqrt{x^{4}+x^{-4}+2}}{x^{3}} d x\) (ii) \(\int \frac{d x}{\sqrt{2 x+3}+\sqrt{2 x-3}} d x\) (iii) \(\int \frac{(\sqrt{x}+1)\left(x^{2}-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}} d x\) (iv) \(\int\left(\frac{1-x^{-2}}{x^{1 / 2}-x^{-1 / 2}}-\frac{2}{x^{3 / 2}}+\frac{x^{-2}-x}{x^{1 / 2}-x^{-1 / 2}}\right) d x\)

Evaluate the following integrals : $$ \int x^{34}\left(1+x^{78}\right)^{1 / 2} d x $$

Evaluate the following integrals : $$ \int x^{-1}\left(1+x^{1 / 3}\right)^{-3} d x $$
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