Answer :

To solve the integral ∫(x^2)/(x^2 + 4) dx, we can use trigonometric substitution. Let x = 2tanθ, and then substitute the expressions for x and dx into the integral. After simplifying and integrating, we obtain the final result.

To solve the integral ∫(x^2)/(x^2 + 4) dx, we can use the trigonometric substitution x = 2tanθ. We choose this substitution because it helps us eliminate the term x^2 + 4 in the denominator.

Using this substitution, we find dx = 2sec^2θ dθ. Substituting x and dx into the integral, we get:

∫((2tanθ)^2)/(4 + (2tanθ)^2) * 2sec^2θ dθ.

Simplifying the expression, we have:

∫(4tan^2θ)/(4 + 4tan^2θ) * 2sec^2θ dθ.

Canceling out the common factors, we get:

∫(2tan^2θ)/(2 + 2tan^2θ) * sec^2θ dθ.

Simplifying further, we have:

∫tan^2θ/(1 + tan^2θ) dθ.

Using the identity 1 + tan^2θ = sec^2θ, we can rewrite the integral as:

∫tan^2θ/sec^2θ dθ.

Simplifying, we get:

∫sin^2θ/cos^2θ dθ.

Using the trigonometric identity sin^2θ = 1 - cos^2θ, we can rewrite the integral as:

∫(1 - cos^2θ)/cos^2θ dθ.

Expanding the integral, we have:

∫(1/cos^2θ) - 1 dθ.

Integrating term by term, we obtain:

∫sec^2θ dθ - ∫dθ.

Integrating sec^2θ gives us tanθ, and integrating dθ gives us θ. Therefore, the final result is:

tanθ - θ + C,

where C is the constant of integration.

So, the solution to the integral ∫(x^2)/(x^2 + 4) dx is tanθ - θ + C, where θ is determined by the substitution x = 2tanθ.

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