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Find the most general antiderivative capital πΉ of π₯ of the function lowercase π of π₯ is equal to π₯ minus three all squared.
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The question gives us a function lowercase π of π₯.
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It wants us to find the most general antiderivative of this function.
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Weβll call this capital πΉ of π₯.
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Remember, an antiderivative means when we differentiate this, we get back to our original function.
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In other words, we want capital πΉ prime of π₯ to be equal to lowercase π of π₯.
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And remember, since the derivative of any constant is equal to zero, we can add any constant we want to our antiderivative.
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And it will still be an antiderivative of our function lowercase π of π₯.
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So we add a constant πΆ to our antiderivative.
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We call this the most general antiderivative, since it will represent all antiderivatives of our function.
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So letβs start trying to find our antiderivative.
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Letβs start by looking at our function lowercase π of π₯.
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We can see itβs π₯ minus three all squared.
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And this is a problem, since this is a composition of functions.
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This can make it much more difficult to find our antiderivatives.
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Instead, letβs simplify our function by distributing the square over our parentheses.
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Weβll distribute our square by using the FOIL method.
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Weβll start by multiplying the first term of each factor together.
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This gives us π₯ times π₯, which is equal to π₯ squared.
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Next, the FOIL method tells us to multiply our two outer terms together.
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This gives us π₯ multiplied by negative three, which is negative three π₯.
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Now, we want to multiply our inner two terms together.
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Again, thatβs negative three times π₯, which is negative three π₯.
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Finally, we want to multiply the last term of each factor together.
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This gives us negative three times negative three, which is equal to nine.
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And we can simplify this, since negative three π₯ minus three π₯ is equal to negative six π₯.
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So now, we can see our function lowercase π of π₯ is a polynomial.
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And we know how to find the antiderivative of each term in a polynomial separately.
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We know to find the antiderivative of π multiplied by π₯ to the πth power, we want to add one to our exponent of π₯ and then divide by this new exponent of π₯.
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This gives us π times π₯ to the power of π plus one divided by π plus one.
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And in the general case, weβll add a constant of integration πΆ.
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Weβll want to do this to each term separately.
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Letβs start with π₯ squared.
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To find an antiderivative of π₯ squared, we can add one to our exponent of two.
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This gives us a new exponent of three.
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But remember, we then need to divide by this new exponent.
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This gives us π₯ cubed over three.
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We now want to find an antiderivative of our second term, negative six π₯.
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One way of doing this is to rewrite our second term as negative six times π₯ to the first power.
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Again, we want to add one to our exponent of π₯.
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This time, itβs equal to one.
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So we get two and then we divide by two.
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So we have negative six π₯ squared divided by two.
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And we know six divided by two is equal to three.
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So weβve shown negative three π₯ squared is an antiderivative of negative six π₯.
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Finally, we want to find an antiderivative of the third term, nine.
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We might be tempted to write this as nine times π₯ to the zeroth power.
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However, thereβs a more simple method of finding an antiderivative in this case.
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We know for any constant πΎ, the derivative of πΎπ₯ with respect to π₯ is just equal to πΎ.
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In other words, for any constant πΎ, πΎπ₯ is an antiderivative of πΎ.
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So when πΎ is equal to nine, we see that nine π₯ is an antiderivative of nine.
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So to find an antiderivative of any constant, we just need to multiply that constant by π₯.
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Remember, though, this is just one possible antiderivative of our function lowercase π of π₯.
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If we add any constant to this, the derivative of that constant is equal to zero.
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So this is still an antiderivative of our function.
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So we add a constant weβve called πΆ to this function.
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This represents all of our possible antiderivatives.
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And this is how we find our most general antiderivative.
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Therefore, we were able to show the most general antiderivative capital πΉ of π₯ of the function lowercase π of π₯ is equal to π₯ minus three all squared is given by capital πΉ of π₯ is equal to π₯ cubed over three minus three π₯ squared plus nine π₯ plus a constant of integration πΆ.