# 8.2.5.evaluate int 64x sec^2(4x) dx

• MHB
• karush
In summary, the conversation discusses how to evaluate the integral $I=\displaystyle \int 64x\sec^2(4x) \, dx$ using integration by parts. The conversation walks through the steps of using this method and ends with the final answer of $I=16x \cdot \tan(4x)+4\ln \left|\cos \left(4x\right)\right|+C$. There is also mention of how the answer differs from that given by Wolfram|Alpha.
karush
Gold Member
MHB
Evaluate $I=\displaystyle \int 64x\sec^2(4x) \, dx$

ok well first $64 \displaystyle\int x\sec^2(4x) \, dx$

off hand not sure what trig id to use or if we need to

Last edited:
integration by parts …

$u = 16x \implies du = 16 \, dx$
$dv = 4\sec^2(4x) \, dx \implies v = \tan(4x)$

$\displaystyle \int 16x \cdot 4\sec^2(4x) \, dx = 16x \cdot \tan(4x) - \int 16\tan(4x) \, dx$

can you finish up?

skeeter said:
integration by parts …

$u = 16x \implies du = 16 \, dx$
$dv = 4\sec^2(4x) \, dx \implies v = \tan(4x)$

$\displaystyle \int 16x \cdot 4\sec^2(4x) \, dx = 16x \cdot \tan(4x) - \int 16\tan(4x) \, dx$

can you finish up?
so if ... (from plug in table)
$\displaystyle 16 \int \tan \left(4x\right)dx =16\left[-\dfrac{1}{4}\ln \left|\cos \left(4x\right)\right|\right]+C =-4\ln \left|\cos \left(4x\right)\right|+C$
Hence
$I=16x \cdot \tan(4x)+4\ln \left|\cos \left(4x\right)\right|+C$

anyway not sure...

isn't W|A answer different or is it??

W/A omitted the absolute value around the log argument.
Why it did, I don't know.

## 1. What does the notation "int" mean in this equation?

The notation "int" stands for integration, which is a mathematical operation that is the inverse of differentiation. It is used to find the area under a curve or the accumulation of a rate of change.

## 2. What is the significance of the number 8.2.5 in this equation?

The number 8.2.5 is the version number of the equation, which indicates that it is the 5th equation in the 8th edition of a particular textbook or reference material. It is used for organization and referencing purposes.

## 3. What does "evaluate" mean in this equation?

In this context, "evaluate" means to find the numerical value of the integral, which involves solving the equation using mathematical techniques.

## 4. What is the variable x representing in this equation?

The variable x represents the independent variable, which is the variable that is being integrated over. In this case, it is being raised to the power of 2 and multiplied by the secant function.

## 5. How is the integral of secant squared (4x) evaluated?

The integral of secant squared (4x) is evaluated using the trigonometric identity, where secant squared (x) is equal to tan(x) + C. In this case, the integral would be equal to tan(4x) + C.

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