Evaluate the Integral integral of (x^2+1)^2 with respect to x

Problem

(∫_^)((x2+1)2*d(x))

Solution

1. Expand the integrand by squaring the binomial (x2+1)

(x2+1)2=(x2)2+2*(x2)*(1)+1

=x4+2*x2+1

2. Rewrite the integral using the expanded polynomial.

(∫_^)((x4+2*x2+1)*d(x))

3. Apply the power rule for integration, (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term individually.

(∫_^)(x4*d(x))=(x5)/5

(∫_^)(2*x2*d(x))=(2*x3)/3

(∫_^)(1*d(x))=x

4. Combine the results and add the constant of integration C

(x5)/5+(2*x3)/3+x+C

Final Answer

(∫_^)((x2+1)2*d(x))=(x5)/5+(2*x3)/3+x+C

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