CALCU-Evaluate the indefinite integral\[(e^5 x-3x^

Solution by Steps

step 1

We start by integrating each term of the given integral separately:

\int \left(e^{5x} - \frac{3}{x^2} + \frac{4}{x}\right) dx

step 2

Integrate the first term:

\int e^{5x} dx = \frac{e^{5x}}{5}

step 3

Integrate the second term:

\int -\frac{3}{x^2} dx = \int -3x^{-2} dx = 3x^{-1} = \frac{3}{x}

step 4

Integrate the third term:

\int \frac{4}{x} dx = 4 \int \frac{1}{x} dx = 4 \ln|x|

step 5

Combine all the integrated terms and add the constant of integration:

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

Answer

\frac{e^{5x}}{5} + \frac{3}{x} + 4 \ln|x| + C

Key Concept

Integration of each term separately

Explanation

To evaluate the indefinite integral, we integrate each term of the given expression separately and then combine the results, adding a constant of integration at the end.

Generated Graph

Solution by Steps

step 1

We start by setting up the definite integral:

step 2

\int_{3\pi}^{5\pi} \cos\left(\frac{x}{2}\right) dx

step 3

To integrate

\cos\left(\frac{x}{2}\right)

, we use the substitution

u = \frac{x}{2}

, hence

du = \frac{1}{2}dx

or

dx = 2du

step 4

Changing the limits of integration: when

x = 3\pi

,

u = \frac{3\pi}{2}

; when

x = 5\pi

,

u = \frac{5\pi}{2}

step 5

The integral becomes:

\int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} 2\cos(u) du

step 6

Integrate

\cos(u)

:

\int 2\cos(u) du = 2\sin(u) + C

step 7

Evaluate the definite integral:

2\sin\left(\frac{5\pi}{2}\right) - 2\sin\left(\frac{3\pi}{2}\right)

step 8

Since

\sin\left(\frac{5\pi}{2}\right) = 1

and

\sin\left(\frac{3\pi}{2}\right) = -1

:

step 9

2(1) - 2(-1) = 2 + 2 = 4

Answer

4

Key Concept

Definite Integral Evaluation

Explanation

The definite integral of

\cos\left(\frac{x}{2}\right)

from

3\pi

to

5\pi

is evaluated by using substitution and then integrating the resulting function. The final result is 4.

Generated Graph

Solution by Steps

step 1

We start by differentiating the given function

f(x) = e^x (x^3 - 3x^2)

step 2

Using the product rule,

\frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v'

, where

u = e^x

and

v = x^3 - 3x^2

step 3

Compute

u' = \frac{d}{dx}[e^x] = e^x

and

v' = \frac{d}{dx}[x^3 - 3x^2] = 3x^2 - 6x

step 4

Applying the product rule:

f'(x) = e^x \cdot (3x^2 - 6x) + e^x \cdot (x^3 - 3x^2)

step 5

Simplify the expression:

f'(x) = e^x (3x^2 - 6x + x^3 - 3x^2) = e^x (x^3 - 6x)

Answer

f'(x) = e^x (x^3 - 6x)

Solution by Steps

step 1

To find the stationary points, we set

f'(x) = 0

step 2

Solve

e^x (x^3 - 6x) = 0

step 3

Since

e^x \neq 0

for all

x

, we solve

x^3 - 6x = 0

step 4

Factor the equation:

x(x^2 - 6) = 0

step 5

Solve for

x

:

x = 0

,

x = \sqrt{6}

, and

x = -\sqrt{6}

Answer

The values of

x

for which the function has a stationary point are

x = 0

,

x = \sqrt{6}

, and

x = -\sqrt{6}

.

Key Concept

Differentiation and Stationary Points

Explanation

To find the derivative of a product of functions, we use the product rule. Stationary points occur where the derivative is zero.

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