Question Statement

Evaluate the indefinite integral: $\int \left(y^2+8y+\sqrt{2}\right)dy$

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Background and Explanation

These problems 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

$\int \left(y^2+8y+\sqrt{2}\right)dy$

We begin by denoting the integral as $I$ and applying the sum rule and constant multiple rule of integration. These properties allow us to break the integral of a sum into the sum of integrals, and to factor constants outside the integral sign.

Let $I=\int \left(y^2+8y+\sqrt{2}\right)dy$

Splitting the integral into three separate terms: $I=\int y^{2}dy+8\int ydy+\sqrt{2}\int 1dy$

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

• For $\int y^{2}dy$: Here $n=2$, so we get $\frac{y^3}{3}$
• For $8\int ydy$: Here $n=1$, so $\int ydy=\frac{y^2}{2}$, giving us $8\cdot \frac{y^2}{2}=4y^2$
• For $\sqrt{2}\int 1dy$: Since $\int 1dy=y$, we get $\sqrt{2}y$

Combining these results and adding the constant of integration $c$ (which represents the family of all possible antiderivatives):

$I=\frac{y^3}{3}+4y^2+\sqrt{2}y+c$

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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.