Evaluate the integral integral ((5x))/(sqrt(1−25x2)) dx and find the exact value of sech( 2).

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:
We need to evaluate the integral $\int \frac{\arcsin(5x)}{\sqrt{1-5x^2}} dx$ .
Assumption: Based on the typical form for integrals involving $\arcsin$ , we assume the denominator is $\sqrt{1-(5x)^2} = \sqrt{1-25x^2}$ . If the denominator is strictly $\sqrt{1-5x^2}$ , the integral would be significantly more complex and not a direct u-substitution.

Use u-substitution.
Let $u = \arcsin(5x)$ .
Find the differential $du$ .
Recall that $\frac{d}{dx}(\arcsin(ax)) = \frac{a}{\sqrt{1-(ax)^2}}$ .
So, differentiating $u$ with respect to $x$ gives:
$du = \frac{5}{\sqrt{1-(5x)^2}} dx = \frac{5}{\sqrt{1-25x^2}} dx$
Rearrange to solve for $\frac{1}{\sqrt{1-25x^2}} dx$ :
$\frac{1}{5} du = \frac{1}{\sqrt{1-25x^2}} dx$
Substitute $u$ and $du$ into the integral.
The integral becomes:
$\int u \cdot \frac{1}{5} du = \frac{1}{5} \int u \, du$
Integrate with respect to $u$ .
$\frac{1}{5} \cdot \frac{u^2}{2} + C = \frac{u^2}{10} + C$
Substitute back $u = \arcsin(5x)$ .
$\frac{(\arcsin(5x))^2}{10} + C$
The final answer is $\boxed{\frac{(\arcsin(5x))^2}{10} + C}$ .

:
We need to find the exact value of $\text{sech}(\ln 2)$ .
Recall the definition of $\text{sech}(x)$ .
$\text{sech}(x) = \frac{1}{\cosh(x)}$
And the definition of $\cosh(x)$ :
$\cosh(x) = \frac{e^x + e^{-x}}{2}$
Substitute $x = \ln 2$ into the definition of $\cosh(x)$ .
$\cosh(\ln 2) = \frac{e^{\ln 2} + e^{-\ln 2}}{2}$
Simplify the exponential terms.
Using the property $e^{\ln a} = a$ :
$e^{\ln 2} = 2$
Using the property $e^{-a} = \frac{1}{e^a}$ :
$e^{-\ln 2} = e^{\ln(2^{-1})} = 2^{-1} = \frac{1}{2}$
Substitute these values back into the expression for $\cosh(\ln 2)$ .
$\cosh(\ln 2) = \frac{2 + \frac{1}{2}}{2}$
Calculate the value of $\cosh(\ln 2)$ .
$\cosh(\ln 2) = \frac{\frac{4}{2} + \frac{1}{2}}{2} = \frac{\frac{5}{2}}{2} = \frac{5}{4}$
Calculate $\text{sech}(\ln 2)$ .
$\text{sech}(\ln 2) = \frac{1}{\cosh(\ln 2)} = \frac{1}{\frac{5}{4}} = \frac{4}{5}$
The final answer is $\boxed{\frac{4}{5}}$ .

Evaluate the integral integral ((5x))/(sqrt(1-25x2)) dx and find the exact value of sech( 2).

Step-by-step solution

Use u-substitution.

Find the differential $du$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = \arcsin(5x)$ .

Recall the definition of $\text{sech}(x)$ .

Substitute $x = \ln 2$ into the definition of $\cosh(x)$ .

Simplify the exponential terms.

Substitute these values back into the expression for $\cosh(\ln 2)$ .

Calculate the value of $\cosh(\ln 2)$ .

Calculate $\text{sech}(\ln 2)$ .

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More Mathematics Questions

Evaluate the integral integral ((5x))/(sqrt(1-25x2)) dx and find the exact value of sech( 2).

Step-by-step solution

Use u-substitution.

Find the differential dududu.

Substitute uuu and dududu into the integral.

Integrate with respect to uuu.

Substitute back u=arcsin⁡(5x)u = \arcsin(5x)u=arcsin(5x).

Recall the definition of sech(x)\text{sech}(x)sech(x).

Substitute x=ln⁡2x = \ln 2x=ln2 into the definition of cosh⁡(x)\cosh(x)cosh(x).