The process of evaluating indefinite integrals is essentially a quest to find the antiderivative, which is a function that, when differentiated, yields the initial function. This process is known as integration and is symbolized as ∫f(x) dx, where f(x) is the function we're integrating. The outcome, also known as the antiderivative, includes a constant of integration, C. This constant is included because the derivative of a constant is zero.
| Topic | Problem | Solution |
|---|---|---|
| None | Evaluate the following indefinite integral. \[ \i… | The integral is in the form of \(\int \frac{a}{\sqrt{x}} dx\), which is a standard integral form. |
| None | Evaluate the integral. \[ \int 9 y^{4} \sqrt{1-9 … | Let's start by identifying a part of the integrand that could be a good choice for a u-substitution… |
| None | Evaluate the indefinite integral. (Remember to us… | The integral is in the form of \(\int \frac{1}{ax+b} dx\), which is a standard integral form. |
| None | Find the particular antiderivative of the followi… | The given derivative is \(C^{\prime}(x)=4 x^{2}-5 x\). To find the antiderivative, we integrate \(C… |
| None | Find the following indefinite integral: \[ \int \… | First, we recognize that this is an integral of the form \(\int u dv\), where \(u = \sqrt{x} = x^{\… |
| None | Find $s(t)$, where $s(t)$ represents the position… | Given the velocity function \(v(t) = 3t^2\), we can find the position function \(s(t)\) by integrat… |
| None | Find $f$ such that $f^{\prime}(x)=4 x^{2}+5 x-5$ … | Given the derivative function $f'(x) = 4x^2 + 5x - 5$ and the initial condition $f(0) = 3$. |
| None | Calculate the indefinite integral \[ \int \frac{4… | We are given the integral \(\int \frac{4 dx}{\sqrt{36-9 x^{2}}}\). |
| None | Find the indefinite integral. \[ \int \frac{x}{\s… | Given the integral \(\int \frac{x}{\sqrt{x-9}} dx\) |
| None | Determine $\int\left[\left(\frac{1}{2}\right)^{x}… | The integral is a sum of two terms, so we can integrate each term separately. |
| None | Find $f$ such that $f^{\prime}(x)=8 x^{2}+5 x-2$ … | We are given that the derivative of the function $f$ is $f^{\prime}(x)=8 x^{2}+5 x-2$ and that the … |
| None | $\int \sin ^{2} x d x$ | Given the integral of \(\sin^2 x\) |
| None | Find $f$ such that $f^{\prime}(x)=x^{2}+2$ and $f… | The problem is asking for a function \(f(x)\) such that its derivative is \(x^{2}+2\) and the funct… |
| None | $\int \cos ^{2}(x+1) d x$ أوجد قِيمة التكامل | \(u = x + 1\) |
| None | $\int \operatorname{tg} x d x=$ | \(\int \operatorname{tg} x dx = \int \frac{\sin x}{\cos x} dx\) |
| None | $\int x^{4} d x$ | Apply the power rule for integration: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) |
| None | [2] \( \int\left(4 \sqrt{x}+\frac{8}{\sqrt{x}}\ri… | \(\int\left(4 \sqrt{x}+\frac{8}{\sqrt{x}}\right) dx = \int(4x^{\frac{1}{2}}+8x^{-\frac{1}{2}}) dx\) |
| None | Find \( \int\left(\frac{3}{\sqrt{x}}\right) d x \) | \int\left(\frac{3}{\sqrt{x}}\right) d x = 3\int\left(x^{-\frac{1}{2}}\right) d x |