Math / Calculus

Questionwhere $C$ is an arbitrary constant.
Find the general antiderivative, $F(x)$ , of the function $f(x)=-\frac{6 \sqrt[2]{x^{7}}}{4}-\frac{6 x^{10}}{7}+8$ $F(x)=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of a function, which means finding a function whose derivative is the given function.
It's like reversing the differentiation process! Watch out! Don't forget to add the "plus C" at the end, representing the arbitrary constant of integration.
This is super important because many different functions can have the same derivative, differing only by a constant.

STEP 2

1. Rewrite the function
2. Integrate term by term
3. Simplify

STEP 3

Let's rewrite our function $f(x)$ to make it easier to integrate.
We can rewrite the square root as a fractional exponent and simplify the fractions: $f(x) = -\frac{6 \sqrt[2]{x^{7}}}{4}-\frac{6 x^{10}}{7}+8 = -\frac{3}{2}x^{\frac{7}{2}} - \frac{6}{7}x^{10} + 8$ This makes each term easier to integrate individually.

STEP 4

Now, let's integrate each term separately using the power rule for integration.
Remember, the power rule says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$ .
We're doing this because the integral of a sum is the sum of the integrals!

STEP 5

For the first term, $-\frac{3}{2}x^{\frac{7}{2}}$ , we have $n = \frac{7}{2}$ , so $n+1 = \frac{7}{2} + 1 = \frac{7}{2} + \frac{2}{2} = \frac{9}{2}$ .
Thus: $\int -\frac{3}{2}x^{\frac{7}{2}} \, dx = -\frac{3}{2} \cdot \frac{x^{\frac{9}{2}}}{\frac{9}{2}} = -\frac{3}{2} \cdot \frac{2}{9} x^{\frac{9}{2}} = -\frac{1}{3}x^{\frac{9}{2}}$ We multiplied by $\frac{2}{9}$ because dividing by $\frac{9}{2}$ is the same as multiplying by its reciprocal.

STEP 6

For the second term, $-\frac{6}{7}x^{10}$ , we have $n = 10$ , so $n+1 = 11$ .
Thus: $\int -\frac{6}{7}x^{10} \, dx = -\frac{6}{7} \cdot \frac{x^{11}}{11} = -\frac{6}{77}x^{11}$

STEP 7

For the constant term, $8$ , remember that it's really $8x^0$ .
So $n = 0$ , and $n+1 = 1$ .
Thus: $\int 8 \, dx = \int 8x^0 \, dx = 8 \cdot \frac{x^1}{1} = 8x$

STEP 8

Putting it all together, we get: $F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$ Remember that C is our constant of integration!

STEP 9

$F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$

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Questionwhere $C$ is an arbitrary constant.
Find the general antiderivative, $F(x)$ , of the function $f(x)=-\frac{6 \sqrt[2]{x^{7}}}{4}-\frac{6 x^{10}}{7}+8$ $F(x)=$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Rewrite the function
2. Integrate term by term
3. Simplify

STEP 3

STEP 4

Now, let's integrate each term separately using the power rule for integration.
Remember, the power rule says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$ .
We're doing this because the integral of a sum is the sum of the integrals!

STEP 5

STEP 6

For the second term, $-\frac{6}{7}x^{10}$ , we have $n = 10$ , so $n+1 = 11$ .
Thus: $\int -\frac{6}{7}x^{10} \, dx = -\frac{6}{7} \cdot \frac{x^{11}}{11} = -\frac{6}{77}x^{11}$

STEP 7

For the constant term, $8$ , remember that it's really $8x^0$ .
So $n = 0$ , and $n+1 = 1$ .
Thus: $\int 8 \, dx = \int 8x^0 \, dx = 8 \cdot \frac{x^1}{1} = 8x$

STEP 8

Putting it all together, we get: $F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$ Remember that C is our constant of integration!

STEP 9

$F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$

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Questionwhere CCC is an arbitrary constant.Find the general antiderivative, F(x)F(x)F(x), of the function f(x)=−6x724−6x107+8f(x)=-\frac{6 \sqrt[2]{x^{7}}}{4}-\frac{6 x^{10}}{7}+8f(x)=−462x7​​−76x10​+8 F(x)=F(x)=F(x)= □\square□

Studdy Solution

STEP 1

STEP 2

1. Rewrite the function2. Integrate term by term3. Simplify

STEP 3

STEP 4

STEP 5

STEP 6

For the **second term**, −67x10-\frac{6}{7}x^{10}−76​x10, we have n=10n = 10n=10, so n+1=11n+1 = 11n+1=11.Thus: ∫−67x10 dx=−67⋅x1111=−677x11\int -\frac{6}{7}x^{10} \, dx = -\frac{6}{7} \cdot \frac{x^{11}}{11} = -\frac{6}{77}x^{11}∫−76​x10dx=−76​⋅11x11​=−776​x11

STEP 7

For the **constant term**, 888, remember that it's really 8x08x^08x0.So n=0n = 0n=0, and n+1=1n+1 = 1n+1=1.Thus: ∫8 dx=∫8x0 dx=8⋅x11=8x\int 8 \, dx = \int 8x^0 \, dx = 8 \cdot \frac{x^1}{1} = 8x∫8dx=∫8x0dx=8⋅1x1​=8x

STEP 8

Putting it all together, we get: F(x)=−13x92−677x11+8x+CF(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + CF(x)=−31​x29​−776​x11+8x+C Remember that *C* is our constant of integration!

STEP 9

F(x)=−13x92−677x11+8x+CF(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + CF(x)=−31​x29​−776​x11+8x+C

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Questionwhere $C$ is an arbitrary constant.
Find the general antiderivative, $F(x)$ , of the function $f(x)=-\frac{6 \sqrt[2]{x^{7}}}{4}-\frac{6 x^{10}}{7}+8$ $F(x)=$ $\square$

1. Rewrite the function
2. Integrate term by term
3. Simplify

For the second term, $-\frac{6}{7}x^{10}$ , we have $n = 10$ , so $n+1 = 11$ .
Thus: $\int -\frac{6}{7}x^{10} \, dx = -\frac{6}{7} \cdot \frac{x^{11}}{11} = -\frac{6}{77}x^{11}$

For the constant term, $8$ , remember that it's really $8x^0$ .
So $n = 0$ , and $n+1 = 1$ .
Thus: $\int 8 \, dx = \int 8x^0 \, dx = 8 \cdot \frac{x^1}{1} = 8x$

Putting it all together, we get: $F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$ Remember that C is our constant of integration!

$F(x) = -\frac{1}{3}x^{\frac{9}{2}} - \frac{6}{77}x^{11} + 8x + C$