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- ∫ 9x+2∫ 9x^(1+1)+ 2x^(0+1)(9/2)x² + 2x
- Looking at this integral I know that I need to be adding to the variables exponents. Starting with the 9x. Even though there is no written number in the exponent, I can write a one in its place. Now I'm going to plus 1 to the exponent and divide just the 9 by the new exponent. Making the anti-derivative of 9x = 9/2 x^2. Then what I would do for the 2 is just add an x because anything raised to the 0 power is . To finally get the anti-derivative of 9/2 x^2 +2x
- For this example we can do it together. For the first term, what will we add to the exponent? That it, 1! So when we do that what should we do next to fully take the integral of the first term? Correct! divide the constant by the new exponent. What is that new exponent? You got it 3. For the next part, after we add one to the exponent, what would we do next? That's it, divide by the new exponent 4. So we get for our final answer, x cubed - x to the fouth.
- ∫ 3x²-4x³x³-x⁴
- Now its your turn. Do this problem on your own! I'll be walking around if you need help
- ∫ (6/2)x³
- Let's take this concept further. Taking the anti-derivative of a square root is easier than you think. We can rewrite the square root of w to w raised to the 1/2 power. Adding one to one half is 3/2 then instead of dividing we can multiply by the reciprocal to get 2/3 w ^3/2. Then we can just add an w to the 1 constant. To get the answer!
- ∫ √w+12/3 w ^3/2 + w
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