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Chapter 7: Problem 1

On which derivative rule is integration by parts based?

Short Answer

Expert verified
Answer: Integration by parts is based on the Product Rule for derivatives which states that the derivative of a product of two functions, u(x) and v(x), is given by: \( (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) \). By rearranging this rule in terms of integration, we get the integration by parts formula: \( \int u'(x)v(x) dx = u(x)v(x) - \int u(x)v'(x) dx \).

Step by step solution

01

Identify the Derivative Rule

Integration by parts is based on the Product Rule for derivatives.
02

Recall the Product Rule

The Product Rule states that the derivative of a product of two functions, say u(x) and v(x), can be given by: \( (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) \).
03

Rearrange the Product Rule to Formulate Integration by Parts

To rearrange the Product Rule in terms of integration, integrate both sides of the equation with respect to x: \( \int(u(x)v(x))' dx = \int(u'(x)v(x) + u(x)v'(x)) dx \). This results in: \( u(x)v(x) = \int u'(x)v(x) dx + \int u(x)v'(x) dx \).
04

Arrive at the Integration by Parts Formula

By rearranging the terms, we get the Integration by Parts formula: \( \int u'(x)v(x) dx = u(x)v(x) - \int u(x)v'(x) dx \). Therefore, the integration by parts is based on the Product Rule for derivatives.

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Most popular questions from this chapter

The Eiffel Tower property Let \(R\) be the region between the curves \(y=e^{-\alpha x}\) and \(y=-e^{-\alpha x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c>0 .\) 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; see the Guided Project The exponential Eiffel Tower.) 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 \(R\).

\(A n\) integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or equivalently \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. \(A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}\) $$\text { Evaluate } \int_{0}^{\pi / 3} \frac{\sin \theta}{1-\sin \theta} d \theta$$

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Determine whether the following statements are true and give an explanation or counterexample. a. The general solution of \(y^{\prime}(t)=20 y\) is \(y=e^{20 t}\). b. The functions \(y=2 e^{-2 t}\) and \(y=10 e^{-2 t}\) do not both satisfy the differential equation \(y^{\prime}+2 y=0\). c. The equation \(y^{\prime}(t)=t y+2 y+2 t+4\) is not separable. d. A solution of \(y^{\prime}(t)=2 \sqrt{y}\) is \(y=(t+1)^{2}\).

Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x=\frac{2 \pi}{3}\)
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