Question Statement

Evaluate the integral:

$\int {\left(4+x^2\right)}^{2}dx$

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Background and Explanation

This problem requires expanding a binomial expression and then applying the power rule for integration. You need to know how to expand $(a+b{)}^2$ and the basic integration formula $\int x^{n}dx$.

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Solution

Let $I=\int {\left(4+x^2\right)}^{2}dx$

First, expand the integrand $(4+x^2{)}^2$ using the algebraic identity $(a+b{)}^2=a^2+2ab+b^2$:

$(4+x^2{)}^2=16+8x^2+x^4$

Now substitute this back into the integral:

$I=\int \left(16+x^4+8x^2\right)dx$

Using the linearity property of integrals, we can split this into three separate integrals:

$I=16\int 1dx+\int x^{4}dx+8\int x^{2}dx$

Apply the power rule for integration $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ to each term:

• $\int 1dx=x$
• $\int x^{4}dx=\frac{x^5}{5}$
• $\int x^{2}dx=\frac{x^3}{3}$

Putting it all together with the constant of integration $c$:

$I=16x+\frac{x^5}{5}+8\cdot \frac{x^3}{3}+c$

Simplifying the final expression:

$I=16x+\frac{x^5}{5}+\frac{8x^3}{3}+c$

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Key Formulas or Methods Used

• Algebraic expansion: $(a+b{)}^2=a^2+2ab+b^2$
• Linearity of integration: $\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx$
• Power rule for integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ (for $n\ne -1$)

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Summary of Steps

1. Expand the squared binomial: $(4+x^2{)}^2=16+8x^2+x^4$
2. Split the integral into three parts using linearity of integration
3. Apply the power rule to integrate each term separately
4. Add the constant of integration $c$
5. Combine terms to write the final answer: $16x+\frac{x^5}{5}+\frac{8x^3}{3}+c$