Work and Integration by Parts

MATH 116 SI worksheet · Fall 2025 · Session 1

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I. Evaluate using integration by parts

  1. \[\displaystyle \int x\cos x\,dx\]
  2. \[\displaystyle \int x\arcsin x\,dx\]
  3. \[\displaystyle \int \arctan x\,dx\]
  4. \[\displaystyle \int x^n\ln x\,dx\]
  5. \[\displaystyle \int e^x\cos x\,dx\]
  6. \[\displaystyle \int x^n e^{x}\,dx\]
  7. \[\displaystyle \int (\ln x)^n\,dx\]
  8. \[\displaystyle \int \cos(\ln x)\,dx\]
  9. \[\displaystyle \int x\ln\left(1+\frac{1}{x}\right) dx\]
  10. \[\displaystyle \int \frac{\sqrt{x^2+1}\,[\ln(x^2+1)-2\ln x]}{x^4}\,dx\]
  11. \[\displaystyle \int \ln\left(\sqrt{1-x}+\sqrt{1+x}\right) dx\]

II. Work problems

  1. Suppose 2 J of work is needed to stretch a spring from its natural length of 30 cm to 42 cm.
    • a. How much work is needed to stretch the spring from 35 cm to 40 cm?
    • b. How far beyond its natural length will a force of 30 N keep the spring stretched?
  2. If 6 J of work is needed to stretch a spring from 10 cm to 12 cm, and another 10 J from 12 cm to 14 cm, what is the natural length of the spring?
  3. A leaky 10 kg bucket is lifted from the ground to a height of 12 m at constant speed with a rope that weighs 0.8 kg/m. The bucket starts with 36 kg of water, which leaks at a constant rate and finishes draining just as the bucket reaches 12 m. How much work is done?
  4. An aquarium 2 m long, 1 m wide and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (The density of water is 1000 kg/m³.)

III. If you have time: extra problems

  1. Two springs with stiffness \(k_1\) and \(k_2\) are connected in series. Find the minimum work needed to stretch the system by \(\Delta L\).
  2. A body of mass \(m\) is pushed up an inclined plane at angle \(\alpha\) with initial velocity \(v_0\). The friction coefficient is \(k\). How far does the body travel before it stops, and how much work does friction do over that distance?
  3. A disc of mass \(m=50\) g slides from rest down a plane inclined at \(\alpha=30^\circ\), then travels \(l=50\) cm along the horizontal before stopping. Find the work done by friction over the whole distance, with friction coefficient \(k=0.15\) on both surfaces.
  4. A chain of mass \(m=0.80\) kg and length \(l=1.5\) m rests on a rough table with one end hanging over the edge. It starts sliding off on its own once the overhanging part is \(l_1=\tfrac{1}{3}\) of its length. How much total work does friction do by the time the chain has slid completely off?
  5. A body of mass \(m\) is lifted from the Earth’s surface by a force that varies with height \(y\) as \(F(y) = 2(ay - 1)mg\), where \(a>0\). Find the work done by this force and the increase in the body’s potential energy over the first half of the ascent.
  6. Let \(P(x)\) be a polynomial with real coefficients. Prove that \(\displaystyle \int_{0}^{\infty} e^{-x}P(x)\,dx=P(0)+P'(0)+P''(0)+\cdots\)
  7. Compute \(\displaystyle \int_{0}^{\pi/4} \tan^{2n}x\,dx\) for \(n\ge1\).