Example Question - constant of integration
Here are examples of questions we've helped users solve.
Finding the Integral of an Exponential Function
<p>\(\int 2 \cdot 5^x dx\)</p> <p>\(= 2 \int 5^x dx\)</p> <p>\(= 2 \cdot \frac{1}{\ln(5)} 5^x + C\)</p> <p>where \(C\) is the constant of integration.</p>
Finding the Integral of a Simple Rational Function
<p>\(\int f(x) \,dx = \int \frac{1}{3x} \,dx\)</p> <p>\(= \frac{1}{3} \int \frac{1}{x} \,dx\)</p> <p>\(= \frac{1}{3} \ln|x| + C\)</p> <p>where \(C\) is the constant of integration.</p>
Integral Calculation for a Rational Function
<p>\( \int f(x) \,dx = \int \frac{3}{x} \,dx \)</p> <p>\( = 3 \int \frac{1}{x} \,dx \)</p> <p>\( = 3 \ln|x| + C \)</p> <p>where \( C \) is the constant of integration.</p>
Integrating a Polynomial Function
Para resolver la integral de la función 3x^2 - 2 con respecto a x, emplearemos las reglas básicas de la integración. La integral de una función de la forma x^n (con n ≠ -1) es (x^(n+1))/(n+1), y la integral de una constante es simplemente la constante multiplicada por x. Así que, aplicando estas reglas, tenemos: ∫(3x^2) dx = 3 * ∫(x^2) dx = 3 * (x^(2+1))/(2+1) = 3 * (x^3)/3 = x^3 Y para la constante: ∫(-2) dx = -2x Por lo tanto, la integral completa es: ∫(3x^2 - 2) dx = x^3 - 2x + C Donde "C" es la constante de integración, que siempre se añade cuando se realiza una integración indefinida.
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
Integral of a Polynomial Function
The integral in the image is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve the integral, integrate each term separately with respect to x: ∫ 8x^3 dx = (8/4)x^4 = 2x^4 ∫ -x^2 dx = -(1/3)x^3 = -x^3/3 ∫ 5x dx = (5/2)x^2 = 5x^2/2 ∫ -1 dx = -x Now, combine the integrated terms: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C is the constant of integration. The final answer for the integral is: 2x^4 - (x^3/3) + (5x^2/2) - x + C
Solving Polynomial Integral with Power Rule
The integral in the image is: ∫(8x^3 - x^2 + 5x - 1)dx To solve this integral, you need to integrate each term separately with respect to x. The integration of a polynomial involves applying the power rule for integration, which says that for any term ax^n, the integral is (a/(n+1))x^(n+1), assuming n ≠ -1. Here's how you integrate each term: For 8x^3: ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 For -x^2: ∫(-x^2) dx = (-1/3)x^(2+1) = -(1/3)x^3 For 5x: ∫5x dx = (5/2)x^(1+1) = (5/2)x^2 For -1: ∫(-1)dx = -x Putting them all together: ∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here C is the constant of integration, which arises because integration is an indefinite integral.
Antiderivative of a Polynomial Function
Certainly! The expression you provided is an integral, and we will find the antiderivative (indefinite integral) of the function. Here's the integral: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, we will integrate each term separately. Recall the power rule for integration, \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where C is the constant of integration. Applying this rule to each term: 1. The integral of \(8x^3\) is \(\frac{8}{4}x^{4} = 2x^4\). 2. The integral of \(-x^2\) is \(-\frac{1}{3}x^{3} = -\frac{1}{3}x^3\). 3. The integral of \(5x\) is \(\frac{5}{2}x^{2} = \frac{5}{2}x^2\). 4. The integral of \(-1\) is \(-x\). Now, putting it all together and adding the constant of integration, \(C\), the antiderivative is: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
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