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University of North Texas

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Problem 72 Easy Difficulty

Verify the formulas by differentiation.
$$\int(3 x+5)^{-2} d x=-\frac{(3 x+5)^{-1}}{3}+C$$

Answer

Take derivative on the right hand side and show we get the expression inside of the integral.

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Video Transcript

they want to verify this formula by using differentiation. So what we want to do is think the derivative of this function on the right hand side and see if we get what we are integrating here. So go ahead and apply are derivative to the right hand side. So derivative of negative three X plus five raised the negative 1/3 plus C. Now the derivative is a linear operator, so we can distribute it across the plus and minus as well as Paul. Any Constance. So we can rewrite this us negative 1/3 and then taking the integral of three x plus five recent negative first power is going to be power and changeable. So power rule, her says. We subtract one from Powers. There's gonna be negative, too now and then we multiply with Clarence and different color negative, too, and we multiply by the whole powers that be negative one and then change Will says we have to take a trip of our inside function, and the derivative of three X plus five would just be three multiplied by three. And it would take the derivative of a constant. See, that's just zero. Let's go ahead and clean this up a little bit. So first thes negatives council out with each other and this three and that three councils out, so we would be left with three X plus five raise to the negative second power. And this is exactly what we were trying to integrate. So this is so that formula up there does hold.