Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int\left(x^{3}-4 x+2\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( x^{3} - 4x + 2 \) is \( 1/4 * x^4 -2x^2 +2x + C \)

Step 1: Apply the power rule to each term

Apply the power rule to each term of the integrand. This means \( \int x^3 dx \) becomes \( 1/4 * x^4 \), \( \int -4x dx \) becomes \( -2x^2 \), and \( \int 2 dx \) becomes \( 2x \).

Step 2: Combine the terms

Combine all the integral terms from step 1 to form the indefinite integral. So, \( \int\left(x^{3}-4 x+2\right) dx \) becomes \( 1/4 * x^4 -2x^2 +2x + C \) where C is the constant of integration.

Step 3: Differentiate the result

Differentiate the result from step 2 to check the solution. The derivative of \( 1/4 * x^4 \) is \( x^3 \), the derivative of \( -2x^2 \) is \( -4x \), and the derivative of \( 2x \) is \( 2 \), and the derivative of a constant is zero. Which gives us back the original integrand, \( x^3 -4x + 2 \).