November 7, 2024
The antiderivative of \( \dfrac{1}{x} \) is a foundational result in calculus, and appears in a variety of contexts. Specifically, it reveals interesting properties of the natural logarithm in calculus. Solution The antiderivative of \( \dfrac{1}{x} \) with respect to \( x \) is: \[\begin{align*} \displaystyle \int \dfrac{1}{x} \, dx = \ln |x| + C […]
November 1, 2024
\[\begin{align*} \boxed{\int 2^x \, dx = \frac{2^x}{\ln(2)} + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: \[\begin{align*} \int 2^x \, dx \end{align*}\] Since \( 2^x \) is an exponential function with a base other than \( e \), to simplify the integration process, we can rewrite \( 2^x \) in […]
November 1, 2024
\[\begin{align*} \boxed{\int \csc(x) \, dx = \ln\left| \tan\left( \dfrac{x}{2} \right) \right| + C} \end{align*}\] where \( C \) is the constant of integration. Introduction: In this article, we will explore how to find the integral of the cosecant function, \( \csc(x) \). The integral of \( \csc(x) \) is less straightforward than those of \( […]
November 1, 2024
\[\begin{align*} \boxed{\int \cot(x) \, dx = \ln|\sin(x)| + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: In this article, we’ll go through the steps to find the integral of \( \cot(x) \). The function \( \cot(x) \) is the ratio of \( \cos(x) \) to \( \sin(x) \), which is […]
November 1, 2024
\[\begin{align*} \boxed{\int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: In this article, we will go through the steps to find the integral of \( \cos(2x) \). We will proceed with $u$-substitution as we know the common integral $\cos x$. \[\begin{align*} \int \cos(2x) […]
November 1, 2024
\[\begin{align*} \boxed{\int e^{2x} \, dx = \dfrac{1}{2} e^{2x} + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: In this article, we’re going to explore how to find the integral of \( e^{2x} \). The function \( e^{2x} \) involves the exponential base \( e \) raised to the power of […]
November 1, 2024
\[\begin{align*} \boxed{\int \sin(2x) \, dx = -\dfrac{1}{2} \cos(2x) + C} \end{align*}\] where \( C \) is the constant of integration. Step-by-Step Solution: To understand why the integral of \( \sin(2x) \) is \( -\dfrac{1}{2} \cos(2x) + C \), let’s use u-substitution. \[\begin{align*} \int \sin(2x) \, dx \end{align*}\] Since we know the common integral $\sin x$, […]
October 29, 2024
Results Summary If you’re searching for “integration of x 2 2,” here are some possible interpretations and their solutions: $$\boxed{\int \frac{x – 2}{2} \, dx = \frac{x^2}{4} – x + C}$$ $$\boxed{\int (x – 2)^2 \, dx = \frac{x^3}{3} – 2x^2 + 4x + C}$$ $$\boxed{\int \frac{x^2}{2} \, dx = \frac{x^3}{6} + C}$$ $$\boxed{\int \frac{x}{2} […]
October 29, 2024
Results Summary If you’re searching for “integration of x 1 x 2,” here are some possible answers: $$\boxed{\int \frac{x + 1}{x + 2} \, dx = \ln |x+2| – \frac{1}{x+2} + C}$$ $$\boxed{\int \frac{x – 1}{x – 2} \, dx = \ln |x – 2| + \frac{1}{x – 2} + C}$$ $$\boxed{\int \frac{x}{1 + x^2} […]
