How to Solve the Definite Integral of 1/(1 + x^2) using Trigonometric Substitution

∫ 1/(1+x^2) dx

The integral you provided is an example of a definite integral

The integral you provided is an example of a definite integral. It involves an algebraic expression with a variable and involves finding the area under a curve. In this case, we have the integral of 1 divided by the quantity 1 plus x squared, with respect to the variable x.

To solve this integral, we can use a technique called trigonometric substitution. First, let’s rewrite our integral as:
∫ dx / (1 + x^2)

Now, let’s substitute x = tanθ. This substitution involves using the trigonometric function tangent.

When we substitute x = tanθ, our integral becomes:
∫ sec^2θ dθ / (1 + tan^2θ)

Using the trigonometric identity: 1 + tan^2θ = sec^2θ, we can simplify the expression further:
∫ dθ / sec^2θ

Using another trigonometric identity: sec^2θ = 1 / cos^2θ, we can simplify it even more:
∫ cos^2θ dθ

Now, we have an integral involving cosine squared, which can be solved using the power-reducing formula. The power-reducing formula states that:
cos^2θ = (1 + cos2θ) / 2

Applying this formula to our integral, we get:
∫ ((1 + cos2θ) / 2) dθ

Using linearity of integration, we can split this into two separate integrals:
(1/2) ∫ (1 + cos2θ) dθ

Now, integrating each term separately:
(1/2) ∫ dθ + (1/2) ∫ cos2θ dθ

The integral of dθ is simply θ. For the second integral, we can use a substitution. Let’s substitute u = 2θ, then du = 2dθ:
(1/2) ∫ dθ + (1/2) ∫ cos(u) (1/2) du

Simplifying further:
(1/2) θ + (1/4) ∫ cos(u) du

Now, integrating cos(u) gives us sin(u):
(1/2) θ + (1/4) sin(u)

Substituting back u = 2θ:
(1/2) θ + (1/4) sin(2θ)

Finally, we have our integral in terms of θ. However, we want to express it in terms of x, which was our original variable of integration. To do that, recall that x = tanθ. Using trigonometric identities, we can express θ in terms of x:
θ = arctan(x)

Therefore, our solution becomes:
(1/2) arctan(x) + (1/4) sin(2 arctan(x))

And that is the final answer to the integral:
∫ 1/(1+x^2) dx = (1/2) arctan(x) + (1/4) sin(2 arctan(x)) + C, where C is the constant of integration.

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