QuestionEvaluate the integral: $\int_{2}\left(\frac{2}{3 x^{2}}-2 \sec ^{2} 5 x\right) d x$

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int_{}\left(\frac{}{3 x^{}}- \sec ^{}5 x\right) d x$ . We need to integrate the function with respect to $x$
3. We will use the basic rules of integration, including the power rule and the integral of $\sec^x$

STEP 2

The given integral is a combination of two different functions. We can separate these into two different integrals.
$\int_{2}\left(\frac{2}{ x^{2}}-2 \sec ^{2}5 x\right) d x = \int_{2}\frac{2}{ x^{2}} dx - \int_{2}2 \sec ^{2}5 x dx$

STEP 3

Now, we will integrate each part separately. For the first part, we will use the power rule of integration, which states that $\int x^n dx = \frac{1}{n+1}x^{n+1}$ .
$\int_{2}\frac{2}{3 x^{2}} dx = \frac{2}{3} \int_{2} x^{-2} dx$

STEP 4

Applying the power rule of integration to the first part, we get $\frac{2}{3} \int_{2} x^{-2} dx = \frac{2}{3} \left[ -x^{-1} \right]_{2}$

STEP 5

For the second part of the integral, we know that the integral of $\sec^2x$ is $\tan x$ . Therefore, we have $\int_{2}2 \sec ^{2}5 x dx =2 \int_{2} \sec ^{2}5 x dx =2 \left[ \frac{1}{5} \tan5x \right]_{2}$

STEP 6

Now, we substitute the limits of integration into both parts of the integral.For the first part, we have $\frac{2}{3} \left[ -x^{-1} \right]_{2} = \frac{2}{3} \left[ -\frac{1}{x} \right]_{2} = \frac{2}{3} \left[ -\frac{1}{2} \right] = -\frac{1}{3}$ For the second part, we have $2 \left[ \frac{1}{5} \tan5x \right]_{2} =2 \left[ \frac{1}{5} \tan10 \right] = \frac{2}{5} \tan10$

STEP 7

Finally, we add the two parts of the integral together to get the final answer.
$-\frac{1}{3} + \frac{2}{5} \tan10$ So, the solution to the integral is $-\frac{1}{3} + \frac{2}{5} \tan10$ .

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