Find ∫(4x^3 + 3)/x^2 dx.

Choose the correct answer from the given options:

1. 4x^2 - 3/x + c
2. 2x^2 + 3/x + c
3. 4x^2 - 1/x^3 + c
4. 2x^2 - 3/x + c

Question

High School

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Answer :

IsabellaRoseDavis IsabellaRoseDavis · Answer 1 · 2024-04-30T13:38:02-04:00

To solve the integral [tex]\int \frac{4x^3 + 3}{x^2} \, dx[/tex], we can simplify the expression inside the integral first.

Rewrite the integrand by dividing each term in the numerator by [tex]x^2[/tex]:

[tex]\int \left( \frac{4x^3}{x^2} + \frac{3}{x^2} \right) \ dx = \int (4x + 3x^{-2}) \, dx[/tex]

Now, the integral can be split into two separate integrals:

[tex]\int 4x \, dx + \int 3x^{-2} \, dx[/tex]

Let's integrate each part separately:

Integrate [tex]4x[/tex]:
- Using the power rule [tex]\int x^n \, dx = \frac{x^{n+1}}{n+1} + C[/tex], we get:
[tex]\int 4x \, dx = 4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2} = 2x^2[/tex]

Integrate [tex]3x^{-2}[/tex]:
- Again using the power rule:
[tex]\int 3x^{-2} \, dx = 3 \cdot \frac{x^{-2+1}}{-2+1} = 3 \cdot \frac{x^{-1}}{-1} = -\frac{3}{x}[/tex]

Putting it all together, the integral is:

[tex]2x^2 - \frac{3}{x} + C[/tex]

Therefore, the answer is option 4: [tex]2x^2 - \frac{3}{x} + C[/tex].