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Chapter 11: Problem 28

Find the indefinite integral and check the result by differentiation. $$ \int \frac{t+2 t^{2}}{\sqrt{t}} d t $$

Short Answer

Expert verified
The indefinite integral of the function \( \frac{t+2 t^{2}}{\sqrt{t}} \) is \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} + C \). The verification by differentiation proves the correctness of this result.

Step by step solution

01

Rewrite the Function

First, rewrite the given function to make it ready for integration. We can rewrite \( \frac{t+2 t^{2}}{\sqrt{t}} \) as \( t^{\frac{1}{2}} + 2t^{\frac{3}{2}} \).
02

Integrate

Apply the rule of integration \(\int x^n dx = \frac{1}{n+1}x^{n+1}\). So, the integral is \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} + C \), where C is the constant of integration.
03

Differentiate the Result

To verify, differentiate the result. The rule of differentiation \(\frac{d}{dx} x^n = nx^{n-1}\) is applied. On differentiating \( \frac{2}{3}t^{\frac{3}{2}} + \frac{4}{5}t^{\frac{5}{2}} \), we get \( \frac{1}{2}t^{-\frac{1}{2}} + 2t^{\frac{3}{2}} \), which simplifies back to the original function, thereby verifying the correctness of the integral.

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Most popular questions from this chapter

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}-x^{3} $$ $$ [0,1] $$

Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20. $$ \int_{0}^{2} \frac{5}{x^{3}+1} d x $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\frac{1}{x}, \quad[1,5] $$

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The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-4}^{0}\left[(x-6)-\left(x^{2}+5 x-6\right)\right] d x $$
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