Math Problem Statement

x2\int x^2(1/(1+16x^2))dx

Solution

We are tasked with evaluating the integral:

x21+16x2dx\int \frac{x^2}{1 + 16x^2} \, dx

Step 1: Simplify the Integral

We can try to simplify this expression. One approach is to separate the integral into a more manageable form by rewriting the numerator x2x^2 and the denominator 1+16x21 + 16x^2 in a way that makes integration simpler.

Step 2: Substitution

A good substitution here is to let:

u=1+16x2u = 1 + 16x^2

Thus, differentiating uu with respect to xx:

du=32xdxdu = 32x \, dx

or equivalently:

du32=xdx\frac{du}{32} = x \, dx

Step 3: Rewrite the Integral

Now, we can rewrite the integral in terms of uu. First, observe that:

x2=u116x^2 = \frac{u - 1}{16}

Substitute this and dx=du32xdx = \frac{du}{32x} into the original integral:

x21+16x2dx=u116udu32x\int \frac{x^2}{1 + 16x^2} \, dx = \int \frac{\frac{u - 1}{16}}{u} \cdot \frac{du}{32x}

This is where the substitution leads to some simplifications.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Substitution Method

Formulas

∫ f(u) du = F(u) + C (indefinite integral)
Substitution: u = 1 + 16x^2

Theorems

Substitution Rule in Integration

Suitable Grade Level

College-Level Calculus