Question

​​​​​​Evaluate the integrals by making appropriate substitutions.

(a) x √{3 x^{2}+4} d x
(b) cos ^{3} x \sin x d x

Answer 1

To evaluate the integral ∫ x √(3x^2+4) dx, use the substitution method. For the integral ∫ cos^3 x sin x dx, also use the substitution method.

Step-by-step explanation:

To evaluate the integral ∫ x √(3x^2+4) dx, we can use the substitution method. Let u = 3x^2+4. Taking the derivative of u with respect to x gives du/dx = 6x, which implies dx = du/(6x). Substituting these expressions into the integral gives: ∫ x √(3x^2+4) dx = ∫ (u-4)/6 √u du. Simplifying and evaluating the integral gives the final result.

For the integral ∫ cos^3 x sin x dx, we can use the substitution method. Let u = cos x, so du = -sin x dx. Substituting these expressions into the integral gives: -∫ u^3 du. Evaluating this integral gives the final result.