Integral Calculus

Exercise 1 : Primitives

Find the primitives of the following functions and the value of the constant:

  1. $f(x) = 3x$ given $F(1) = 2$
  2. $f(x) = 2x + 3$ given $F(1) = 0$

Exercise 2 : Definite Integrals (Cosine)

Evaluate the following definite integrals:

  1. $\int_{0}^{\pi/2} 3 \cos x , dx$
  2. $\int_{-\pi/2}^{\pi/2} 3 \cos x , dx$
  3. $\int_{0}^{\pi} 3 \cos x , dx$

Exercise 3 : Absolute Areas

Obtain the absolute values of the areas corresponding to the following integrals:

  1. $\int_{-2}^{0} (x - 2) , dx$
  2. $\int_{0}^{2} (x - 2) , dx$
  3. $\int_{0}^{4} (x - 2) , dx$

Exercise 4 : Integrate and Verify

Integrate and verify the result by differentiating:

  1. $\int \frac{2 , dx}{(x + 1)^2} = \frac{x - 1}{x + 1} + C$
  2. $2 \int \sin^2 (4x - 1) , dx = x - \frac{1}{8} \sin(8x - 2) + C$
  3. $\int \frac{1 - x^2}{(1 + x^2)^2} , dx = \frac{x}{1 + x^2} + C$

Exercise 5 : Standard Integrals (Table)

Evaluate the following integrals by using the table of standard integrals:

  1. $\int \frac{dx}{x - a}$
  2. $\int \frac{1}{\cos^2 x} , dx$
  3. $\int \frac{a}{\sqrt{x^2 + a^2}} , dx$
  4. $\int \sin^2 \alpha , d\alpha$
  5. $\int a^t , dt$
  6. $\int \sqrt[3]{x^7} , dx$
  7. $\int 5(x^2 + x^3) , dx$
  8. $\int \left(\frac{3}{2}t^3 + 4t\right) , dt$

Exercise 6 : Integration by Parts and Reduction

Integrate by parts the following integrals:

  1. $\int x \ln x , dx$
  2. $\int x^2 \cos x , dx$
  3. $\int x^2 \ln x , dx$
  4. $\int x^2 \cosh \frac{x}{a} , dx$
  5. Find the reduction formula for $\int \cos^n x , dx$ ($n \neq 0$)
  6. Find the general formula for $\int x^n \ln x , dx$

Exercise 7 : Substitution Techniques

Use a suitable substitution to evaluate the following integrals:

  1. $\int \sin(\pi x) , dx$
  2. $\int 3e^{3x-6} , dx$
  3. $\int \frac{dx}{2x + a}$
  4. $\int (ax + b)^5 , dx$

Exercise 8 : Trig and Rational Integrals

Evaluate the following integrals:

  1. $\int \cot 2x , dx$
  2. $\int \frac{2x}{a + x^2} , dx$
  3. $\int \frac{x^{39}}{x^{40} + 21} , dx$
  4. $\int \frac{\sinh u}{\cosh^2 u} , du$

Exercise 9 : Composite Integrals

Evaluate the following integrals:

  1. $\int (\sin^4 x + 8 \sin^3 x + \sin x) \cos x , dx$
  2. $\int x^4 \sqrt{3x^5 - 1} , dx$
  3. $\int \frac{-x}{\sqrt{a - x^2}} , dx$
  4. $\int x \cos x^2 , dx$

Exercise 10 : Mixed Problems

Mixed questions:

  1. $\int \frac{e^x}{e^x + 1} , dx$
  2. $\int \cos \left(x - \frac{\pi}{2}\right) , dx$
  3. $\int \cos^3 x , dx$
  4. $\int \frac{1}{x \ln x} , dx$
  5. $\int \frac{3x^2 - 1}{x^3 - x} , dx$
  6. $\int \frac{1}{(1 + x^2) \tan^{-1} x} , dx$

Exercise 11 : Partial Fractions

Using partial fractions, integrate the following functions:

  1. $\int \frac{1}{2 - x - x^2} , dx$
  2. $\int \frac{2x + 3}{x(x - 1)(x + 2)} , dx$
  3. $\int \frac{x^2}{(x - 1)(x - 2)(x - 3)} , dx$
  4. $\int \frac{x}{x^4 - x^2 - 2} , dx$
  5. $\int \frac{1}{x^3 + 3x^2 - 4} , dx$
  6. $\int \frac{x^2 - 1}{x^4 + x^2 + 1} , dx$
  7. $\int \frac{x}{(x - 1)(x^2 + 2x + 5)} , dx$

Exercise 12 : Definite Integrals — Practice

Evaluate the following definite integrals:

  1. $\int_{-2}^{2} (x^5 - 8x^3 + x + 7) , dx$
  2. $\int_{0}^{1} \frac{1}{1 + x} , dx$
  3. $\int_{0}^{2} \sin t , dt$
  4. $3 \int_{100}^{125} dt$

Exercise 13 : Areas Between Curves

Find the value of the absolute area between the following boundary lines:

  1. $y = x^3$; $x$-axis; $a = \frac{1}{2}$; $b = 2$
  2. $y = \cos x$; $x$-axis; $a = -\frac{3\pi}{2}$; $b = \frac{5\pi}{6}$
  3. What is the value of the area between the curves $y = 4x^3$ and $y = 6x^2 - 2$? (Hint: Sketch the graphs of both functions first. Note that for $x = 1$ both curves have a point in common, but do not intersect.)

Exercise 14 : Improper Integrals

Integrate the following:

  1. $\int_{4}^{\infty} \frac{d\rho}{\rho^2}$
  2. $\int_{10}^{\infty} \frac{dx}{x}$
  3. $\int_{r_0}^{\infty} \frac{dr}{r^2}$
  4. $\int_{1}^{\infty} \frac{d\lambda}{\lambda}$
  5. $\int_{1}^{\infty} \frac{dr}{r^3}$
  6. $\int_{1}^{\infty} \left(1 + \frac{1}{x^2}\right) dx$
  7. $\int_{-\infty}^{-1} \frac{dx}{x^2}$
  8. $\int_{1}^{\infty} \frac{1}{\sqrt{x}} , dx$

Exercise 15 : Work — Constant Force

A force in a conservative field is given by $\mathbf{F} = (2, 6, 1) , \text{N}$. A body is moved along the line given by $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{i}$ from point $\mathbf{r}(0) = \mathbf{r}_0$ to point $\mathbf{r}(2) = \mathbf{r}_0 + 2\mathbf{i}$. Calculate the work done.

Exercise 16 : Work — Variable Force

A force in a conservative field is given by $\mathbf{F} = (x, y, z) , \text{N}$. A body moves from the origin of the coordinate system to the point $P = (5, 0, 0)$. Calculate the work done.

Exercise 17 : Trigonometric Substitution

Use trigonometric substitutions to evaluate the following integrals:

  1. $\displaystyle \int \frac{dx}{\sqrt{a^2 - x^2}}$
  2. $\displaystyle \int \frac{x^2}{\sqrt{x^2 + a^2}},dx$
  3. $\displaystyle \int \frac{dx}{x^2\sqrt{x^2 - a^2}}$

Exercise 18 : Numerical Integration (Trapezoid and Simpson)

Approximate the following integrals numerically and compare methods:

  1. Approximate $\displaystyle \int_{0}^{1} e^{-x^2} ;dx$ using the trapezoidal rule with $n=4$.
  2. Approximate the same integral using Simpson’s rule with $n=4$.
  3. Give a brief error estimate for both methods (order of magnitude suffices).

Exercise 19 : Arc Length and Surface Area

Compute arc lengths and surface areas:

  1. Find the arc length of $y=\ln(\cos x)$ for $x\in[0,\pi/4]$.
  2. Find the surface area generated by rotating $y=\sqrt{x}$ about the $x$-axis for $x\in[0,1]$.

Exercise 20 : Centers of Mass

Find centroids and moments for planar regions and rods:

  1. Find the centroid (center of area) of the region bounded by $y=x^2$ and $y=2x$ for $x\in[0,2]$.
  2. A thin rod occupies $[0,1]$ with linear density $\rho(x)=x$. Find the mass and the center of mass.

Exercise 21 : Polar Coordinates and Areas

Evaluate areas and integrals in polar form:

  1. Find the area enclosed by the cardioid $r=1+\cos\theta$.
  2. Find the area enclosed by $r=a\sin(2\theta)$ for the appropriate interval of $\theta$.

Exercise 22 : Convergence of Improper Integrals

Determine convergence or divergence (justify your test):

  1. $\displaystyle \int_{1}^{\infty} \frac{1}{x^{p}},dx$ (state for which $p$ it converges).
  2. $\displaystyle \int_{0}^{1} \frac{1}{x^{p}},dx$ (state for which $p$ it converges).

Exercise 23 : Beta and Gamma — Basics

Short exercises on special integrals:

  1. Show that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ using the substitution $x=t^2$.
  2. Evaluate $B(3,2)$ and express it in terms of factorials.

Exercise 24 : Applications to Probability

Compute expectations and variances using integrals:

  1. For an exponential distribution with parameter $\lambda>0$, compute $E[X]=\int_{0}^{\infty} \lambda x e^{-\lambda x} ,dx$.
  2. Compute $\operatorname{Var}(X)$ for the same distribution.