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Chapter 4: Problem 18

Find the indefinite integral and check the result by differentiation. $$ \int(1+3 t) t^{2} d t $$

Short Answer

Expert verified
The indefinite integral of \(x^{2}+\frac{1}{(3 x)^{2}}\) is \(\frac{x^{3}}{3} -\frac{1}{9x} + C\).

Step by step solution

01

Identify the elements to integrate individually

The function \(x^{2}+\frac{1}{(3 x)^{2}}\) can be broken down into two parts: \(x^{2}\) and \(\frac{1}{(3 x)^{2}}\). Both of these parts should be handled separately.
02

Apply the power rule to \(x^{2}\)

The power rule states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\). This gives us the integral of \(\int x^{2} dx = \frac{x^{3}}{3}\).
03

Transform \(\frac{1}{(3x)^{2}}\) into a suitable form

Rewrite \(\frac{1}{(3x)^{2}}\) as \(\frac{1}{9} * \frac{1}{x^{2}} = \frac{1}{9x^{2}}\). The constant \(\frac{1}{9}\) can be moved in front of the integral.
04

Apply the power rule to \(\frac{1}{9x^{2}}\)

Applying the power rule again to \(\frac{1}{9x^{2}}\) gives \(\int \frac{1}{9x^{2}} dx = -\frac{1}{9x}\).
05

Combine results

Adding these two results gives the indefinite integral: \(\frac{x^{3}}{3} -\frac{1}{9x} + C\), where \(C\) is the constant of integration.
06

Check the result by differentiation

The derivative of \(\frac{x^{3}}{3} -\frac{1}{9x} + C\) is \(x^{2} + \frac{1}{(3 x)^{2}}\), which confirms that the integral was calculated correctly.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([a, b]\), then \(f\) is integrable on \([a, b]\).

Prove that \(\frac{d}{d x}\left[\int_{u(x)}^{v(x)} f(t) d t\right]=f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)\).

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{3}} \sin t^{2} d t $$

Evaluate the integral in terms of (a) natural logarithms and (b) inverse hyperbolic functions. \(\int_{-1 / 2}^{1 / 2} \frac{d x}{1-x^{2}}\)

Find the derivative of the function. \(y=\sinh ^{-1}(\tan x)\)
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