November 1, 2024

Understanding the Integral of \( 2^x \)

\[\begin{align*} \boxed{\int 2^x \, dx = \frac{2^x}{\ln(2)} + C} \end{align*}\] 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…

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Understanding the Derivative of \( 2^x \)

\[\begin{align*} \boxed{\frac{d}{dx} \left( 2^x \right) = 2^x \ln(2)} \end{align*}\] Introduction: Finding the derivative of \( 2^x \) helps us understand how the function changes as \( x \) changes. Unlike derivatives of polynomials or basic trigonometric functions, differentiating an exponential function involves the natural logarithm. Let’s walk through the steps. Step-by-Step Solution: To differentiate \(…

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Understanding the Integral of \( \csc(x) \)

\[\begin{align*} \boxed{\int \csc(x) \, dx = \ln\left| \tan\left( \dfrac{x}{2} \right) \right| + C} \end{align*}\] 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 \(…

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Understanding the Integral of \( \cot(x) \)

\[\begin{align*} \boxed{\int \cot(x) \, dx = \ln|\sin(x)| + C} \end{align*}\] 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…

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Understanding the Integral of \( \cos(2x) \)

\[\begin{align*} \boxed{\int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) + C} \end{align*}\] 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)…

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Understanding the Integral of \( e^{2x} \)

\[\begin{align*} \boxed{\int e^{2x} \, dx = \dfrac{1}{2} e^{2x} + C} \end{align*}\] 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…

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Understanding the Integral of \( \sin(2x) \)

\[\begin{align*} \boxed{\int \sin(2x) \, dx = -\dfrac{1}{2} \cos(2x) + C} \end{align*}\] 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 $\sin x$,…

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