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karush
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$\int x (5^{-x^2})dx= -\frac{1}{2} \int 5^{-x^{2}}(-2x)dx$
how is $\frac{1}{2}$ in front of the $\int$ derived
how is $\frac{1}{2}$ in front of the $\int$ derived
Last edited:
karush said:$\int x (5^{-x^2})dx= -\frac{1}{2} \int 5^{-x^{2}}(-2x)dx$
how is $\frac{1}{2}$ in front of the $\int$ derived
An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and x is a variable. The variable, x, is typically in the exponent, giving rise to the name "exponential".
To integrate an exponential function, you can use the power rule of integration, which states that the integral of x^n is (x^(n+1))/(n+1), where n is any real number. For an exponential function, this means that the integral of ab^x is (ab^x)/ln(b) + C, where C is the constant of integration.
Integrating exponential functions can be used to model various natural phenomena, such as population growth, radioactive decay, and compound interest. It is also commonly used in physics, chemistry, and economics to describe rates of change.
One special rule for integrating exponential functions is the logarithmic integration rule, which states that the integral of e^x is ln(e^x) + C = x + C. This rule can be helpful in simplifying integrals of exponential functions with a base of e.
Yes, exponential functions can be integrated using substitution. However, it may not always be the most efficient method. In some cases, using the power rule or logarithmic rule may be easier. It is important to consider the form of the function and choose the most appropriate integration technique.