Simplify The Following: ∫ ( 2 X + 10 − 2 X ) D X \int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) Dx ∫ ( X ​ 2 ​ + 1 0 − 2 X ) D X

Introduction

In this article, we will delve into the world of calculus and focus on simplifying a given integral. The integral in question is $\int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) dx$. Our goal is to break down the integral into manageable parts, apply various integration techniques, and ultimately simplify the expression.

Breaking Down the Integral

The given integral consists of two terms: $\frac{2}{\sqrt{x}}$ and $10^{-2x}$. To simplify the integral, we will tackle each term separately and then combine the results.

Simplifying the First Term

The first term, $\frac{2}{\sqrt{x}}$, can be rewritten as $2x^{-\frac{1}{2}}$. This is a power function with a negative exponent, which can be integrated using the power rule of integration.

The Power Rule of Integration

The power rule of integration states that if $f(x) = x^n$, then $\int f(x) dx = \frac{x^{n+1}}{n+1} + C$. In our case, $n = -\frac{1}{2}$, so we can apply the power rule to integrate the first term.

\int 2x^{-\frac{1}{2}} dx = \frac{2x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} + C
= \frac{2x^{\frac{1}{2}}}{\frac{1}{2}} + C
= 4x^{\frac{1}{2}} + C

Simplifying the Second Term

The second term, $10^{-2x}$, is an exponential function with a negative exponent. We can integrate this term using the general rule for integrating exponential functions.

The General Rule for Integrating Exponential Functions

If $f(x) = e^{ax}$, then $\int f(x) dx = \frac{e^{ax}}{a} + C$. In our case, $a = -2$, so we can apply the general rule to integrate the second term.

\int 10^{-2x} dx = \frac{10^{-2x}}{-2} + C
= -\frac{1}{2} \cdot 10^{-2x} + C

Combining the Results

Now that we have simplified each term, we can combine the results to obtain the final expression for the integral.

\int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) dx = 4x^{\frac{1}{2}} - \frac{1}{2} \cdot 10^{-2x} + C

Conclusion

In this article, we simplified the given integral $\int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) dx$ by breaking it down into manageable parts and applying various integration techniques. We used the power rule of integration to simplify the first term and the general rule for integrating exponential functions to simplify the second term. The final expression for the integral is $4x^{\frac{1}{2}} - \frac{1}{2} \cdot 10^{-2x} + C$.

Final Answer

The final answer is $\boxed{4x^{\frac{1}{2}} - \frac{1}{2} \cdot 10^{-2x} + C}$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Related Articles

• Simplifying the Integral: $\int\left(\frac{1}{x}+e^x\right) dx$
• Simplifying the Integral: $\int\left(x^2+e^{-x}\right) dx$
  Simplify the Given Integral: A Comprehensive Approach - Q&A ===========================================================

Introduction

In our previous article, we simplified the given integral $\int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) dx$ by breaking it down into manageable parts and applying various integration techniques. In this article, we will address some common questions and concerns related to the simplification of the integral.

Q&A

Q: What is the power rule of integration?

A: The power rule of integration states that if $f(x) = x^n$, then $\int f(x) dx = \frac{x^{n+1}}{n+1} + C$. This rule is used to integrate power functions with positive exponents.

Q: How do I apply the power rule of integration?

A: To apply the power rule of integration, simply substitute the value of $n$ into the formula and simplify. For example, if $f(x) = x^3$, then $\int f(x) dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C$.

Q: What is the general rule for integrating exponential functions?

A: The general rule for integrating exponential functions states that if $f(x) = e^{ax}$, then $\int f(x) dx = \frac{e^{ax}}{a} + C$. This rule is used to integrate exponential functions with positive exponents.

Q: How do I apply the general rule for integrating exponential functions?

A: To apply the general rule for integrating exponential functions, simply substitute the value of $a$ into the formula and simplify. For example, if $f(x) = e^{2x}$, then $\int f(x) dx = \frac{e^{2x}}{2} + C$.

Q: Can I use the power rule of integration to simplify the second term of the given integral?

A: No, the power rule of integration is only applicable to power functions with positive exponents. The second term of the given integral, $10^{-2x}$, is an exponential function with a negative exponent, so the power rule of integration cannot be used to simplify it.

Q: Can I use the general rule for integrating exponential functions to simplify the first term of the given integral?

A: No, the general rule for integrating exponential functions is only applicable to exponential functions with positive exponents. The first term of the given integral, $\frac{2}{\sqrt{x}}$, is a power function with a negative exponent, so the general rule for integrating exponential functions cannot be used to simplify it.

Q: What is the final expression for the integral?

A: The final expression for the integral is $4x^{\frac{1}{2}} - \frac{1}{2} \cdot 10^{-2x} + C$.

Q: What is the value of $C$ in the final expression for the integral?

A: The value of $C$ is a constant of integration, which can be any real number. The value of $C$ is determined by the specific problem being solved and is not a fixed value.

Conclusion

In this article, we addressed some common questions and concerns related to the simplification of the given integral $\int\left(\frac{2}{\sqrt{x}}+10^{-2x}\right) dx$. We provided explanations and examples to help clarify the concepts and techniques used in the simplification of the integral.

Final Answer

The final answer is $\boxed{4x^{\frac{1}{2}} - \frac{1}{2} \cdot 10^{-2x} + C}$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Related Articles

• Simplifying the Integral: $\int\left(\frac{1}{x}+e^x\right) dx$
• Simplifying the Integral: $\int\left(x^2+e^{-x}\right) dx$