Question Statement

$\int \left(\sqrt{y}+\frac{1}{y^2}\right)dy$

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Background and Explanation

This problem involve finding indefinite integrals of algebraic functions. You'll need to apply the linearity property of integrals (which allows you to integrate term by term and pull out constants) and the power rule for integration. Remember that integration is the reverse process of differentiation, so we're looking for functions whose derivatives give us the integrands.

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Solution

$\int \left(\sqrt{y}+\frac{1}{y^2}\right)dy$

First, we rewrite the integrand using exponent notation to make the power rule applicable. Recall that $\sqrt{y}=y^{1/2}$ and $\frac{1}{y^2}=y^{-2}$.

Let $I=\int \left(\sqrt{y}+\frac{1}{y^2}\right)dy$

Rewriting and splitting the integral: $I=\int y^{1/2}dy+\int y^{-2}dy$

Applying the power rule $\int y^{n}dy=\frac{y^{n+1}}{n+1}$ to each term:

First term: $\int y^{1/2}dy$ $=\frac{y^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\frac{y^{\frac{3}{2}}}{\frac{3}{2}}=\frac{2}{3}{y}^{\frac{3}{2}}$

Second term: $\int y^{-2}dy$ $=\frac{y^{-2+1}}{-2+1}=\frac{y^{-1}}{-1}=-\frac{1}{y}$

Combining both results with the constant of integration:

$I=\frac{2}{3}{y}^{\frac{3}{2}}-\frac{1}{y}+c$

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Key Formulas or Methods Used

• Linearity of Integration: $\int [f(y)+g(y)]dy=\int f(y)dy+\int g(y)dy$ and $\int k\cdot f(y)dy=k\int f(y)dy$ (where $k$ is a constant)
• Power Rule: $\int y^{n}dy=\frac{y^{n+1}}{n+1}+c$ for $n\ne -1$
• Exponent Conversion: $\sqrt[n]{y^m}=y^{m/n}$ and $\frac{1}{y^n}=y^{-n}$

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Summary of Steps

1. Identify the structure: Check if the integral contains sums, constants, or terms that need algebraic rewriting (like radicals or reciprocals).
2. Apply linearity: Split the integral into separate terms and pull out constant coefficients.
3. Convert to power form: Rewrite any roots as fractional exponents ($\sqrt{y}=y^{1/2}$) and reciprocals as negative exponents ($\frac{1}{y^2}=y^{-2}$).
4. Integrate term by term: Apply the power rule $\frac{y^{n+1}}{n+1}$ to each term, being careful with arithmetic on fractions.
5. Simplify coefficients: Reduce any numerical fractions (e.g., $8\times \frac{1}{2}=4$) and rewrite negative exponents as fractions if desired.
6. Add the constant: Don't forget $+c$ at the end, as this is an indefinite integral representing a family of functions.