Final answer:
By using a trigonometric substitution of x = tan(theta), the integral becomes a simpler expression in terms of theta, which is then solved for an antiderivative and reversed back to x to evaluate the integral.
Step-by-step explanation:
To evaluate the integral ∫₁√(2)1/(1+x²)³ / 2 dx using trigonometric substitution, we start by making the substitution x = tan(theta), which comes from the trigonometric identity 1 + tan²(theta) = sec²(theta). The differential dx will be sec²(theta) d(theta). After substituting, we change the limits of integration to match the new variable (theta). By simplifying and evaluating the new expression, we eventually find the antiderivative in terms of theta. After computing the antiderivative, we need to reverse the substitution to express the result back in terms of x and finally evaluate the integral between the new limits.