IBHardCalculus10–12

A Definite Integral Cleared by Substitution

Um Integral Definido Resolvido por Substituição

Evaluate $\int_{0}^{2} 2x\sqrt{x^{2} + 1}\,dx$ , giving an exact value and a decimal to three significant figures.

Look for an inner function whose derivative is already sitting in the integrand as a factor. When you substitute in a definite integral, the limits belong to the new variable too — changing them saves you converting back.

Solution

Step 1 of 7

1.Let $u = x^{2} + 1$ , so $\frac{du}{dx} = 2x$ and $du = 2x\,dx$ — exactly the factor already present.