Math / Calculus

QuestionQuestion Evaluate the indefinite integral given below. $\int\left(\frac{-6 x^{6}-4 x}{x^{2}}-4 e^{6 x}\right) d x$
Provide your answer below: $\int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of a function that's a mix of a polynomial divided by $x^2$ and an exponential term.
Basically, we're looking for a function whose derivative is the one given to us! Watch out! Don't forget the "+C" for indefinite integrals – it represents that family of functions that all have the same derivative!
Also, be careful with the signs when simplifying the polynomial fraction.

STEP 2

1. Simplify the Integrand
2. Integrate Term by Term
3. Combine and Simplify

STEP 3

Let's rewrite the integrand to make it easier to integrate.
We can split the fraction and simplify each term: $\frac{-6x^6 - 4x}{x^2} = \frac{-6x^6}{x^2} + \frac{-4x}{x^2} = -6x^4 - \frac{4}{x}$ So, our integral becomes: $\int \left( -6x^4 - \frac{4}{x} - 4e^{6x} \right) dx$ This is much easier to work with now!

STEP 4

Now, we can integrate each term separately.
Remember, the power rule for integration is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$ ).
Also, $\int \frac{1}{x} dx = \ln|x| + C$ and $\int e^{ax} dx = \frac{1}{a}e^{ax} + C$ .

STEP 5

Let's tackle the first term: $\int -6x^4 dx = -6 \cdot \frac{x^5}{5} = -\frac{6}{5}x^5 + C_1$

STEP 6

Next, the second term: $\int -\frac{4}{x} dx = -4 \int \frac{1}{x} dx = -4 \ln|x| + C_2$

STEP 7

Finally, the exponential term: $\int -4e^{6x} dx = -4 \cdot \frac{1}{6} e^{6x} = -\frac{2}{3}e^{6x} + C_3$

STEP 8

Let's put it all together.
We can combine the constants of integration ( $C_1$ , $C_2$ , and $C_3$ ) into a single constant, $C$ . $\int \left( -6x^4 - \frac{4}{x} - 4e^{6x} \right) dx = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C$

STEP 9

$\int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C$

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QuestionQuestion Evaluate the indefinite integral given below. $\int\left(\frac{-6 x^{6}-4 x}{x^{2}}-4 e^{6 x}\right) d x$
Provide your answer below: $\int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x=$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Simplify the Integrand
2. Integrate Term by Term
3. Combine and Simplify

STEP 3

STEP 4

Now, we can integrate each term separately.
Remember, the power rule for integration is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$ ).
Also, $\int \frac{1}{x} dx = \ln|x| + C$ and $\int e^{ax} dx = \frac{1}{a}e^{ax} + C$ .

STEP 5

Let's tackle the first term: $\int -6x^4 dx = -6 \cdot \frac{x^5}{5} = -\frac{6}{5}x^5 + C_1$

STEP 6

Next, the second term: $\int -\frac{4}{x} dx = -4 \int \frac{1}{x} dx = -4 \ln|x| + C_2$

STEP 7

Finally, the exponential term: $\int -4e^{6x} dx = -4 \cdot \frac{1}{6} e^{6x} = -\frac{2}{3}e^{6x} + C_3$

STEP 8

Let's put it all together.
We can combine the constants of integration ( $C_1$ , $C_2$ , and $C_3$ ) into a single constant, $C$ . $\int \left( -6x^4 - \frac{4}{x} - 4e^{6x} \right) dx = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C$

STEP 9

$\int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C$

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Studdy Solution

STEP 1

STEP 2

1. Simplify the Integrand2. Integrate Term by Term3. Combine and Simplify

STEP 3

STEP 4

STEP 5

Let's **tackle** the first term: ∫−6x4dx=−6⋅x55=−65x5+C1 \int -6x^4 dx = -6 \cdot \frac{x^5}{5} = -\frac{6}{5}x^5 + C_1 ∫−6x4dx=−6⋅5x5​=−56​x5+C1​

STEP 6

Next, the second term: ∫−4xdx=−4∫1xdx=−4ln⁡∣x∣+C2 \int -\frac{4}{x} dx = -4 \int \frac{1}{x} dx = -4 \ln|x| + C_2 ∫−x4​dx=−4∫x1​dx=−4ln∣x∣+C2​

STEP 7

Finally, the exponential term: ∫−4e6xdx=−4⋅16e6x=−23e6x+C3 \int -4e^{6x} dx = -4 \cdot \frac{1}{6} e^{6x} = -\frac{2}{3}e^{6x} + C_3 ∫−4e6xdx=−4⋅61​e6x=−32​e6x+C3​

STEP 8

STEP 9

∫(−4e6x+−6x6−4xx2)dx=−65x5−4ln⁡∣x∣−23e6x+C \int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C ∫(−4e6x+x2−6x6−4x​)dx=−56​x5−4ln∣x∣−32​e6x+C

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1. Simplify the Integrand
2. Integrate Term by Term
3. Combine and Simplify

Let's tackle the first term: $\int -6x^4 dx = -6 \cdot \frac{x^5}{5} = -\frac{6}{5}x^5 + C_1$

Next, the second term: $\int -\frac{4}{x} dx = -4 \int \frac{1}{x} dx = -4 \ln|x| + C_2$

Finally, the exponential term: $\int -4e^{6x} dx = -4 \cdot \frac{1}{6} e^{6x} = -\frac{2}{3}e^{6x} + C_3$

$\int\left(-4 e^{6 x}+\frac{-6 x^{6}-4 x}{x^{2}}\right) d x = -\frac{6}{5}x^5 - 4\ln|x| - \frac{2}{3}e^{6x} + C$