Question Video: Finding the Integration of an Exponential Function | Nagwa Question Video: Finding the Integration of an Exponential Function | Nagwa

# Question Video: Finding the Integration of an Exponential Function Mathematics

Determine β« 4πe^(3π₯) ππ₯.

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### Video Transcript

Determine the integral of four ππ to the power of three π₯.

To integrate four ππ to the power of three π₯, we actually have a rule. And the rule weβre gonna use tells us that the integral of π to the power of ππ₯ plus π is equal to one over π π to the power of ππ₯ plus π plus π. And this is true when π is not equal to zero. And this comes from something that youβd have recovered previously which is that if you differentiate π to the power of ππ₯ plus π, then the result is ππ to the power of ππ₯ plus π. And that relationship was found using the chain rule.

Okay, so weβve now got a general rule for integrating π to the power of ππ₯ plus π which we have derived from the derivative of π to the power of ππ₯ plus π. Things to watch out for here, donβt forget the π because obviously, we need to add π as this is our constant of integration. And just to remind you why we need that constant of integration, weβre gonna look at π¦ equals four π₯ squared plus three. If I actually differentiate four π₯ squared plus three, then the derivative is gonna be equal to eight π₯. However, if I wanted to use integration to actually find the original function, so what π¦ was equal to, then what I would have to do is integrate eight π₯. And then, what I get if I use the rule for integration would be eight π₯ to the power of one plus one, cause you add one to the exponent, and then divided by the new exponent, which would give me eight π₯ squared over two, which would give me four π₯ squared.

So if we look back at our original function, okay great. Yep, weβve got our four π₯ squared. And actually, we can see that we havenβt got the positive three. So therefore, we have to add π, so add our constant of integration. And thatβs because if weβre actually working to find our function, so find π¦, we donβt know if there was an additional number added on to the original function. So if weβd had four π₯ squared plus nine and four π₯ squared minus 12, that still will have differentiated to eight π₯. So thatβs why we need the constant of integration, so that we could say that we know something could be added to the term that weβve already got.

Okay, so now we know what we need to do. Letβs use it to actually determine the integral of our expression. So first, well weβre gonna take a look. Weβve actually got four π in front of our π to the power of three π₯. Well, this doesnβt actually change the integration. So this coefficient wonβt affect the integration. So we can carry on and integrate. So if we look at the example weβve got, so we look at our rule, we can see that our π is gonna be equal to three. Okay, so now letβs use that rule and integrate our expression.

So when we have actually differentiated four ππ to the power of three π₯, weβre gonna get four π over three π to the power of three π₯ and then plus π. And thatβs because if we look back at our general rule, we can see that itβs gonna be one over π. Well, in this case, itβs gonna be the coefficient, which is four π, over our π, which is three. And then the π to the power of three π₯ remains unchanged. And then we have to remember to add the constant of integration. So we get our final answer that if we integrate four ππ to the power of three π₯, the result is four π over three π to the power of three π₯ plus π.