An integral is a mathematical concept that represents the area under a curve on a graph. It is used to calculate the total value of a function over a given interval.
Solving integrals is important because it allows us to find the exact value of a function over a given interval, which can provide valuable information in many fields such as physics, engineering, and economics.
The process for solving an integral involves applying specific rules and techniques, such as substitution, integration by parts, or partial fractions, to manipulate the integrand into a form that can be easily integrated. Once the integrand is in a simpler form, the integral can be solved using standard integration formulas.
To solve this integral, we can use the substitution method by letting u= 4-x. This will change the integral to ∫ (-u+4)/u du, which can be broken up into two separate integrals: ∫ -du/u + ∫ 4/u du. The first integral can be solved using the power rule, while the second integral can be solved using the natural log rule. Once both integrals are solved, we can substitute the value of u back in to get the final answer.
Some tips for solving integrals include identifying patterns in the integrand, simplifying the expression by factoring or using trigonometric identities, and practicing regularly to become familiar with different integration techniques. It is also helpful to check your answer by differentiating it and making sure it matches the original function.