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Math  /  Calculus

QuestionFind the general antiderivative of f(x)=−6x5+5x+5x4+5xf(x)=-6 x^{5}+\frac{5}{x}+\frac{5}{x^{4}}+5 \sqrt{x}

Studdy Solution

STEP 1

What is this asking? We need to find the *most general* antiderivative of the given function f(x)f(x), which means finding a function whose derivative is f(x)f(x), and don't forget the "*plus C*"! Watch out! Remember the power rule for integration, and how it works differently for 1x\frac{1}{x}.
Also, don't forget to rewrite roots as fractional powers before integrating.

STEP 2

1. Rewrite the function
2. Integrate each term
3. Simplify and add the constant of integration

STEP 3

Let's **rewrite** f(x)f(x) to make it easier to integrate.
We can rewrite 5x\frac{5}{x} as 5⋅1x5 \cdot \frac{1}{x}, 5x4\frac{5}{x^{4}} as 5x−45x^{-4}, and 5x5\sqrt{x} as 5x125x^{\frac{1}{2}}.
So, our function becomes: f(x)=−6x5+5⋅1x+5x−4+5x12f(x) = -6x^5 + 5 \cdot \frac{1}{x} + 5x^{-4} + 5x^{\frac{1}{2}} This makes it much clearer how to apply the power rule in the next step!

STEP 4

The power rule for integration says ∫xn dx=xn+1n+1\int x^n \, dx = \frac{x^{n+1}}{n+1}, where n≠−1n \neq -1.
Applying this to −6x5-6x^5, we get: ∫−6x5 dx=−6⋅x5+15+1=−6⋅x66=−x6\int -6x^5 \, dx = -6 \cdot \frac{x^{5+1}}{5+1} = -6 \cdot \frac{x^6}{6} = -x^6

STEP 5

Remember, ∫1x dx=ln⁡∣x∣\int \frac{1}{x} \, dx = \ln|x|.
So, ∫5⋅1x dx=5∫1x dx=5ln⁡∣x∣\int 5 \cdot \frac{1}{x} \, dx = 5\int \frac{1}{x} \, dx = 5\ln|x|

STEP 6

Using the power rule again: ∫5x−4 dx=5⋅x−4+1−4+1=5⋅x−3−3=−53x−3=−53x3\int 5x^{-4} \, dx = 5 \cdot \frac{x^{-4+1}}{-4+1} = 5 \cdot \frac{x^{-3}}{-3} = -\frac{5}{3}x^{-3} = -\frac{5}{3x^3}

STEP 7

One more time with the power rule: ∫5x12 dx=5⋅x12+112+1=5⋅x3232=5⋅23x32=103x32\int 5x^{\frac{1}{2}} \, dx = 5 \cdot \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} = 5 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = 5 \cdot \frac{2}{3}x^{\frac{3}{2}} = \frac{10}{3}x^{\frac{3}{2}}

STEP 8

Now, we **add** the results of our integrations: −x6+5ln⁡∣x∣−53x3+103x32-x^6 + 5\ln|x| - \frac{5}{3x^3} + \frac{10}{3}x^{\frac{3}{2}}

STEP 9

Since the derivative of a constant is always zero, we need to add a constant of integration, usually denoted as "*C*".
This gives us the **general antiderivative**: −x6+5ln⁡∣x∣−53x3+103x32+C-x^6 + 5\ln|x| - \frac{5}{3x^3} + \frac{10}{3}x^{\frac{3}{2}} + C

STEP 10

The general antiderivative of f(x)f(x) is −x6+5ln⁡∣x∣−53x3+103x32+C-x^6 + 5\ln|x| - \frac{5}{3x^3} + \frac{10}{3}x^{\frac{3}{2}} + C.

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