Example Question - indefinite integral
Here are examples of questions we've helped users solve.
Solving an Improper Integral
<p>لحل التكامل الوارد في الصورة، أولاً، نبسط المقام كالتالي:</p> <p>\[ \int \frac{1}{x^{4} \sqrt{9 - x^{2}}} dx \]</p> <p>نستخدم التعويض التالي:</p> <p>\[ x = 3\sin(\theta) \]</p> <p>\[ dx = 3\cos(\theta) d\theta \]</p> <p>\[ 9 - x^{2} = 9 - 9\sin^{2}(\theta) = 9\cos^{2}(\theta) \]</p> <p>نعوض في التكامل الأصلي:</p> <p>\[ = \int \frac{3\cos(\theta)}{(3\sin(\theta))^{4} \sqrt{9\cos^{2}(\theta)}} d\theta \]</p> <p>\[ = \frac{1}{27} \int \frac{\cos(\theta)}{\sin^{4}(\theta)\cos(\theta)} d\theta \]</p> <p>\[ = \frac{1}{27} \int \csc^{4}(\theta) d\theta \]</p> <p>\[ = \frac{1}{27} \int \csc^{2}(\theta) \csc^{2}(\theta) d\theta \]</p> <p>نستخدم التكامل بالتجزئة أو جداول التكامل لإيجاد:</p> <p>\[ \int \csc^{2}(\theta) d\theta = - \cot(\theta) \]</p> <p>\[ \int \csc^{4}(\theta) d\theta = -\frac{1}{3} \cot(\theta)\csc^{2}(\theta) - \frac{2}{3} \int \csc^{2}(\theta) d\theta \]</p> <p>نحل التكامل ونعيد التعبير عن الناتج بدلالة \( x \) بعد العودة إلى التعويض الأول.</p> <p>هذا التكامل قد يتطلب تقنيات متقدمة في التعامل مع التكاملات الغير مناسبة والدوال المثلثية، ولكن من المهم العودة إلى التعويض الأصلي للوصول إلى النتيجة النهائية بدلالة \( x \).</p>
Integral of a Rational Function over a Radical
<p>\(\int \frac{1}{x^4 \sqrt{9 - x^2}} dx\)</p> <p>لحل هذا التكامل، نقوم عادةً بتعيين \(x = 3\sin(\theta)\) لتبسيط المعادلة تحت الجذر.</p> <p>إذًا، \(dx = 3\cos(\theta) d\theta\) و \(9 - x^2 = 9 - 9\sin^2(\theta)\).</p> <p>نستخدم هوية الجيب وجيب التمام \(1 - \sin^2(\theta) = \cos^2(\theta)\), لنحصل على:</p> <p>\(\sqrt{9 - x^2} = \sqrt{9\cos^2(\theta)} = 3|\cos(\theta)|\).</p> <p>وبما أن \(\theta\) هو الأركسين لـ \(x/3\), فإن \(\cos(\theta) \geq 0\) وبالتالي \(|\cos(\theta)| = \cos(\theta)\).</p> <p>التكامل يصبح:</p> <p>\(\int \frac{1}{(3\sin(\theta))^4 \cdot 3\cos(\theta)} \cdot 3\cos(\theta) d\theta\)</p> <p>\(\int \frac{1}{(3\sin(\theta))^4} d\theta\)</p> <p>\(\int \frac{1}{81\sin^4(\theta)} d\theta\)</p> <p>لتبسيط التعبير ن更 استخدم الهوية \(sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\):</p> <p>\(\int \frac{4}{81(1 - \cos(2\theta))^2} d\theta\)</p> <p>ثم نستخدم تبسيطًا شائعًا ونقدم التكامل بصيغة تحويلات إلى كثيرات حدود:</p> <p>\(\int \frac{4}{81(1 - u)^2} \frac{du}{-2}\) حيث \(u = \cos(2\theta)\)</p> <p>\(-\frac{8}{81} \int \frac{1}{(1 - u)^2} du\)</p> <p>\(-\frac{8}{81} \left(-\frac{1}{1 - u}\right) + C\)</p> <p>\(\frac{8}{81(1 - \cos(2\theta))} + C\)</p> <p>ثم نعيد التعبير الى المتغير \(x\) باستخدام الهويات الأصلية:</p> <p>\(\frac{8}{81\left(1 - \left(1 - 2\sin^2\left(\frac{x}{3}\right)\right)\right)} + C\)</p> <p>\(\frac{8}{81\left(2\sin^2\left(\frac{x}{3}\right)\right)} + C\)</p> <p>\(\frac{4}{81\sin^2\left(\frac{x}{3}\right)} + C\)</p> <p>وهذا هو الحل النهائي للتكامل المعطى.</p>
Finding the Integral of an Exponential Function
<p>\int 7^x dx = \int e^{x \ln(7)} dx</p> <p>Let u = x \ln(7) \Rightarrow du = \ln(7) dx</p> <p>dx = \frac{du}{\ln(7)}</p> <p>\int e^{x \ln(7)} dx = \int e^u \frac{du}{\ln(7)}</p> <p>\frac{1}{\ln(7)}\int e^u du = \frac{1}{\ln(7)} e^u + C</p> <p>\frac{1}{\ln(7)} e^{x \ln(7)} + C = \frac{7^x}{\ln(7)} + C</p> <p>So, the integral of f(x) = 7^x is \frac{7^x}{\ln(7)} + C</p>
Solving an Exponential Function's Integral
<p>\[\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\]</p> <p>where \(C\) is the constant of integration.</p>
Finding the Integral of a Function Involving a Fraction with a Variable in the Denominator
<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>
Evaluation of an Improper Integral
<p>The image shows an improper integral of the form \(\int \sqrt{x} dx\). To solve this integral, we use the power rule for integration. The power rule states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), for any real number \(n\) not equal to -1, where \(C\) is the constant of integration.</p> <p>The integral can thus be solved as follows:</p> <p>\begin{align*} \int \sqrt{x} dx &= \int x^{1/2} dx \\ &= \frac{x^{(1/2) + 1}}{(1/2) + 1} + C \\ &= \frac{x^{3/2}}{3/2} + C \\ &= \frac{2}{3} x^{3/2} + C. \end{align*}</p> <p>Therefore, the solution to the integral is \(\frac{2}{3} x^{3/2} + C\).</p>
Integration of a Power Function
<p>\(\int x^{-2} dx\)</p> <p>\(= \int x^{-2+1}(-2+1)^{-1} dx\)</p> <p>\(= \int x^{-1}(-1)^{-1} dx\)</p> <p>\(= -x^{-1+1}(-1+1)^{-1} + C\)</p> <p>\(= -1 \cdot x^0 + C\)</p> <p>\(= -1 + C\)</p> <p>\(= -1 + C\)</p>
Integrating a Polynomial Function
Claro, podemos resolver la integral de la función f(x) = 3x^2 - 2 con respecto a x. La integral indefinida se obtiene al encontrar la antiderivada de f(x). En este caso, necesitamos aplicar la regla de potencias para la integración, que dice que ∫x^n dx = (x^(n+1))/(n+1) + C, donde C es la constante de integración. Vamos a integrar la función término por término: ∫(3x^2 - 2) dx = 3∫x^2 dx - ∫2 dx Ahora aplicaremos la regla de potencias mencionada anteriormente: Para 3∫x^2 dx, n es igual a 2, así que la antiderivada será (x^(2+1))/(2+1), que se simplifica a (x^3)/3. Para ∫2 dx, simplemente tratamos 2 como una constante multiplicando a x^0, por lo que su antiderivada es 2x (dado que la antiderivada de x^0 es x). Por lo tanto: 3∫x^2 dx = 3 * (x^3)/3 = x^3 (la constante 3 se cancela con el denominador 3). ∫2 dx = 2x Sumamos las antiderivadas para obtener la integral indefinida completa: ∫(3x^2 - 2) dx = x^3 - 2x + C Donde C es la constante de integración que no se conoce a menos que se den más condiciones o límites para la integral.
Integration Using Trigonometric Substitution
Para resolver la integral que está en la imagen, podemos utilizar una sustitución trigonométrica debido a la presencia de una raíz cuadrada de una diferencia de cuadrados. La expresión bajo la raíz cuadrada, \( 1-x^2 \), sugiere usar la identidad trigonométrica \( \sin^2(\theta) + \cos^2(\theta) = 1 \), por lo que podemos hacer la sustitución \( x = \sin(\theta) \), \( dx = \cos(\theta) d\theta \). Entonces, la expresión bajo la raíz cuadrada se convierte en \( \cos^2(\theta) \), cuya raíz cuadrada es simplemente \( \cos(\theta) \). La integral original se transforma así: \[ \int \frac{x^2}{\sqrt{1 - x^2}} dx \] Sustituimos \( x = \sin(\theta) \) y \( dx = \cos(\theta) d\theta \): \[ \int \frac{\sin^2(\theta)}{\cos(\theta)} \cos(\theta) d\theta = \int \sin^2(\theta) d\theta \] Ahora podemos utilizar la identidad trigonométrica para \( \sin^2(\theta) \): \[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \] Entonces la integral se convierte en: \[ \int \frac{1 - \cos(2\theta)}{2} d\theta = \frac{1}{2} \int (1 - \cos(2\theta)) d\theta \] La cual se puede integrar término por término: \[ \frac{1}{2} \left( \int 1 d\theta - \int \cos(2\theta) d\theta \right) = \frac{1}{2} (\theta - \frac{\sin(2\theta)}{2}) + C \] Donde \( C \) es la constante de integración. Finalmente, debemos volver a expresar \( \theta \) en términos de \( x \) utilizando la sustitución original \( x = \sin(\theta) \). Para \( \theta \), usamos \( \theta = \arcsin(x) \) y para \( \sin(2\theta) \), usamos la identidad \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2x\sqrt{1 - x^2} \). Por lo tanto, la integral original se convierte en: \[ \frac{1}{2} \left( \arcsin(x) - \frac{1}{2} \cdot 2x\sqrt{1 - x^2} \right) + C \] O simplificando: \[ \frac{\arcsin(x)}{2} - \frac{x\sqrt{1 - x^2}}{2} + C \] Esta es la expresión de la integral indefinida que ha pedido calcular.
Integrating a Complicated Function by Parts
To solve the integral provided in your image, we can separate it into two integrals and solve each one individually. Here is the integral: ∫(√(2x) - 3 + 7x^(3/4) + (2 / (3x + 1))) dx Now we can separate it into individual integrals: ∫√(2x) dx - ∫3 dx + ∫7x^(3/4) dx + ∫(2 / (3x + 1)) dx Let's integrate each term one by one. 1. For ∫√(2x) dx, let u = 2x. Then, du = 2 dx, or (1/2)du = dx. The integral becomes (1/2)∫u^(1/2) du, which evaluates to (1/2) * (2/3)u^(3/2) + C = (1/3) * (2x)^(3/2) + C1. 2. For ∫3 dx, the integral is simply 3x + C2. 3. For ∫7x^(3/4) dx, the integral is 7 * (4/7)x^(7/4) + C = 4x^(7/4) + C3. 4. For ∫(2 / (3x + 1)) dx, let u = 3x + 1. Then, du = 3 dx, or (1/3)du = dx. The integral becomes (2/3)∫(1/u) du, which evaluates to (2/3)ln|u| + C = (2/3)ln|3x + 1| + C4. Now combine all the results back together to get the final solution: (1/3) * (2x)^(3/2) + 3x + 4x^(7/4) + (2/3)ln|3x + 1| + C Where C is the constant of integration that is the sum of all individual constants C1, C2, C3, and C4.
Integral of Polynomial Function
The integral in the image is an indefinite integral of a polynomial function. To solve the integral, you would integrate each term separately, applying the power rule for integration. Here's the integration term-by-term: ∫(8x^3 - x^2 + 5x - 1) dx = ∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx Now applying the power rule for integration (which states that ∫x^n dx = x^(n+1)/(n+1) for all n ≠ -1), we get: = 8 * x^(3+1)/(3+1) - x^(2+1)/(2+1) + 5 * x^(1+1)/(1+1) - x + C = 8 * x^4/4 - x^3/3 + 5 * x^2/2 - x + C Simplifying, we have: = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C So the integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C represents the constant of integration.
Solving Indefinite Integral of Polynomial Expression
The image shows the integral of a polynomial expression. I'll guide you through the steps to solve the indefinite integral: ∫(8x^3 - x^2 + 5x - 1)dx To integrate this polynomial term by term, you would use the power rule for integration. The power rule states that the integral of x^n dx is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration. Applying this rule to each term, you would get: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (1/4) * x^4 = 2x^4 ∫(-x^2) dx = (-1) * ∫x^2 dx = (-1) * (1/3) * x^3 = -1/3x^3 ∫5x dx = 5 * ∫x dx = 5 * (1/2) * x^2 = 5/2x^2 ∫(-1) dx = -1 * ∫dx = -1 * x = -x Adding them all together, you get: 2x^4 - 1/3x^3 + 5/2x^2 - x + C Therefore, the integral of the given expression is: 2x^4 - 1/3x^3 + 5/2x^2 - x + C
Integration using Power Rule
The image shows an integral expression that you'd like to evaluate. The integral is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve the integral, we use the power rule of integration which states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n is a real number different from -1, and C is the constant of integration. Let's integrate each term individually: ∫ 8x^3 dx = (8x^(3+1))/(3+1) = (8x^4)/4 = 2x^4 ∫ -x^2 dx = -(x^(2+1))/(2+1) = -(x^3)/3 ∫ 5x dx = (5x^(1+1))/(1+1) = (5x^2)/2 ∫ -1 dx = -1x Putting it all together, the indefinite integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C is the constant of integration.
Integral of a Polynomial Expression
Certainly! The given expression is an integral that you need to evaluate. You have the following expression: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, you integrate each term separately with respect to x. The integral of a sum or difference of functions is the sum or difference of their integrals. Here's how to integrate each term: 1. ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 2. ∫-x^2 dx = -(1/3)x^(2+1) = -(1/3)x^3 3. ∫5x dx = (5/2)x^(1+1) = (5/2)x^2 4. ∫-1 dx = -x Now combine these results to get the complete antiderivative: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C represents the constant of integration, which is a standard addition in indefinite integrals, as there are an infinite number of antiderivatives differing by a constant.
Indefinite Integral of a Polynomial
To solve the given indefinite integral, integrate each term separately with respect to x. The integral you provided is: ∫(8x^3 - x^2 + 5x - 1) dx Apply the power rule of integration (which states that ∫x^n dx = (x^(n+1)) / (n+1) + C for any n that is not equal to -1) to each term: ∫8x^3 dx = (8/4)x^(3+1) + C_1 = 2x^4 + C_1 ∫-x^2 dx = (-1/3)x^(2+1) + C_2 = -(1/3)x^3 + C_2 ∫5x dx = (5/2)x^(1+1) + C_3 = (5/2)x^2 + C_3 ∫-1 dx = -x + C_4 Combine these integrals and their constants into one expression: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C is the constant of integration that combines C_1, C_2, C_3, and C_4 into a single constant, since any finite number of constants can be combined into one. So the integral of the expression is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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