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Chapter 7: Problem 9

Evaluate the following integrals. $$\int t e^{t} d t$$

Short Answer

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Question: Evaluate the integral 鈭玹 * e^t dt. Answer: 鈭玹 * e^t dt = t * e^t - e^t + C.

Step by step solution

01

Identify u and dv

We have to choose what parts of the integrand will be 'u' and 'dv'. In this case, let u be the algebraic factor and dv be the exponential factor: $$u = t$$ $$dv = e^t dt$$ Now, we need to find the derivatives and integrals of these parts.
02

Calculate du and v

We'll find the derivative of u and the integral of dv: $$du = \frac{d(u)}{dt} dt = dt$$ $$v = \int dv = \int e^t dt = e^t + C_1$$ Notice that we don't need the constant \(C_1\) since it will not affect our final result (it will be included in the final constant). Now we have: $$u = t$$ $$du = dt$$ $$v=e^t$$ $$dv=e^t dt$$
03

Apply the integration by parts formula

Now we can use the integration by parts formula, which states that: $$\int u \ dv = uv - \int v \ du$$ Plugging in the values, we get: $$\int t \ e^t \ dt = t e^t - \int e^t \ dt$$
04

Evaluate the remaining integral

Now, we need to evaluate the integral of the remaining term: $$\int e^t \ dt = e^t + C_2$$
05

Substitute the result and simplification

Substituting this back into our equation from step 3, we have: $$\int t \ e^t \ dt = t e^t - (e^t + C_2)$$ Simplifying the expression: $$\int t \ e^t \ dt = t e^t - e^t + C$$ where \(C = -C_2\) is the constant of integration. And that's the final result: $$\int t \ e^t \ dt = t e^t - e^t + C$$

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Most popular questions from this chapter

Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 2 and an illustration.

Let \(R\) be the region between the curves \(y=e^{-c x}\) and \(y=-e^{-c x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c \geq 0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves.) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R\). Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0.\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\) (Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique \(332(2004): 571-584 .)\)

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{\left(e^{x}+e^{-x}\right)^{2}}$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt{1+\sqrt{x}}} ; x=\left(u^{2}-1\right)^{2}$$

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int \frac{4 x^{6}}{x^{4}+3 x^{2}} d x\), the first step is to find the partial fraction decomposition of the integrand. b. The easiest way to evaluate \(\int \frac{6 x+1}{3 x^{2}+x} d x\) is with a partial fraction decomposition of the integrand. c. The rational function \(f(x)=\frac{1}{x^{2}-13 x+42}\) has an irreducible quadratic denominator. d. The rational function \(f(x)=\frac{1}{x^{2}-13 x+43}\) has an irreducible quadratic denominator.
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