We can generalize this by multiplying both sides of the equation by a constant □: Multiplying the constant of integration by a constant yields a constant, so we have We want to take out the factor of □ + 1 from the integral and divide both sides of the equation by □ + 1 however, Substituting these expressions into the reverse chain rule, we obtain We can recall that the power rule for differentiation tells us that □ ′ ( □ ) = ( □ + 1 ) □ . Let □ ( □ ) = □ for some unknown constant □ and let □ ( □ ) be aĭifferentiable function. There are many applications of the chain rule however, in this explainer, we will focus on two specific applications of this result. If □ is differentiable at □ and □ is differentiable at □ ( □ ), This is known as the reverse chain rule since it is found by reversing the chain rule by integration. This is now in the form of an integral result, where we need to add a constant of integration as usual: We can reverse this process by integrating both sides of this result with respect to □. Sense, a very useful application of this property is that any derivative result can be stated as an integration result by reversing the process.įor example, we recall that the chain rule tells us that if □ is differentiable at □ and □ isĭ d □ ( □ ( □ ( □ ) ) ) = □ ′ ( □ ) □ ′ ( □ ( □ ) ). Furthermore, we can verify our answer to an integration problem by differentiating it. This means that knowing how to perform differentiation on certain expressionsĬan help solve certain types of integration problems. Integration and differentiation are the reverse processes of each other. In this explainer, we will learn how to evaluate integrals of functions in the form
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