Example Question - indefinite integration
Here are examples of questions we've helped users solve.
Indefinite Integration of a Constant Divided by a Variable
<p>\int \frac{3}{x} \, dx = 3 \int \frac{1}{x} \, dx</p> <p>3 \int \frac{1}{x} \, dx = 3 \ln|x| + C</p> <p>\therefore \int \frac{3}{x} \, dx = 3 \ln|x| + C</p>
Indefinite Integration of a Negative Power Function
<p>The general formula for finding the indefinite integral of a power function \(\int x^n dx\) is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).</p> <p>Here, \(n = -2\). Applying the formula, we get:</p> <p>\(\int x^{-2} dx = \frac{x^{-2+1}}{-2+1} + C\)</p> <p>\(\int x^{-2} dx = \frac{x^{-1}}{-1} + C\)</p> <p>\(\int x^{-2} dx = -x^{-1} + C\)</p> <p>\(\int x^{-2} dx = -\frac{1}{x} + C\)</p>
Indefinite Integration of Tan(x)dx within Specified Bounds
The image shows a mathematical problem asking to solve the definite integral of tan(x)dx within the bounds from 0 to pi. However, the image is blurry and I cannot completely confirm the integral bounds. Please confirm or clarify the bounds of the integral so I can provide the correct solution. If the bounds of the integral are indeed from 0 to pi, we run into an issue. Since tan(x) has a vertical asymptote at x = pi/2, where the function becomes undefined (tan(x) approaches positive or negative infinity as x approaches pi/2 from the left or right, respectively), the integral of tan(x) from 0 to pi does not converge to a finite number. Thus, the integral is not defined in the classical sense, but one could approach it in the context of Cauchy principal values or other extended frameworks. Please verify the bounds so that we could proceed with the appropriate solution if necessary.
Solving Integration Problems
The given image presents three separate integration problems, asking to integrate the following expressions with respect to x: (a) ∫(2x - 3)dx (b) ∫(2 + 3x)dx (c) ∫(x^2 - 1)dx Let's integrate each one: (a) To integrate ∫(2x - 3)dx, we integrate each term separately: ∫2x dx = 2∫x dx = 2 * (1/2)x^2 = x^2 ∫(-3) dx = -3x Therefore, ∫(2x - 3)dx = x^2 - 3x + C, where C is the constant of integration. (b) Similarly for ∫(2 + 3x)dx: ∫2 dx = 2x ∫3x dx = 3 * (1/2)x^2 = (3/2)x^2 Adding these together gives us ∫(2 + 3x)dx = 2x + (3/2)x^2 + C. (c) For ∫(x^2 - 1)dx: ∫x^2 dx = (1/3)x^3 ∫(-1) dx = -x Combining these results: ∫(x^2 - 1)dx = (1/3)x^3 - x + C. So, the integrals for each part are: (a) x^2 - 3x + C (b) 2x + (3/2)x^2 + C (c) (1/3)x^3 - x + C
Integration of Polynomial Functions
The image contains a mathematical expression that represents an integral problem to be solved. Here is the integral from the image: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, we will integrate each term separately using the power rule of integration, which states that for any term ax^n, the integral is (a/(n+1))x^(n+1), provided n is not equal to -1. Here are the steps: 1. Integrate \(8x^3\): \[ \int 8x^3 \, dx = \frac{8}{4}x^{3+1} = 2x^4 \] 2. Integrate \(-x^2\): \[ \int -x^2 \, dx = \frac{-1}{3}x^{2+1} = -\frac{1}{3}x^3 \] 3. Integrate \(5x\): \[ \int 5x \, dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2 \] 4. Integrate \(-1\): \[ \int -1 \, dx = -x \] Now, let's put all these integrated terms together: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] Where \(C\) is the constant of integration, which appears because we are performing indefinite integration. So, the final answer is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
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