Techniques for Integration: Get Stronger

Substitution

  1. Why is uu-substitution referred to as change of variable?
  2. If f=gh, when reversing the chain rule, ddx(gh)(x)=g(h(x))h(x), should you take u=g(x) or u=h(x)?

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form f(u)du.

  1. For x>1x2x1dx=215x1(3x2+4x+8)+C;u=x1
  2. x4x2+9dx=144x2+9+C;u=4x2+9
  3. (x+1)4dx;u=x+1
  4. (2x3)7dx;u=2x3
  5. xx2+1dx;u=x2+1
  6. (x1)(x22x)3dx;u=x22x
  7. cos3θdθ;u=sinθ

In the following exercises, use a suitable change of variables to determine the indefinite integral.

  1. x(1x)99dx
  2. (11x7)3dx
  3. cos3θsinθdθ
  4. cos2(πt)sin(πt)dt
  5. tsin(t2)cos(t2)dt
  6. x2(x33)2dx
  7. y5(1y3)3/2dy
  8. (1cos3θ)10cos2θsinθdθ
  9. (sin2θ2sinθ)(sin3θ3sin2θ)3cosθdθ

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.

  1. y=x(1x2)3 over [1,2]
  2. y=x(x2+1)2 over [1,1]

In the following exercises, use a change of variables to evaluate the definite integral.

  1. 10x1+x2dx
  2. 10t21+t3dt
  3. π/40sinθcos4θdθ

In the following exercises, evaluate the indefinite integral f(x)dx with constant C=0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=xaf(t)dt, with a the left endpoint of the given interval.

  1. cos(ln(2x))xdx on [0,2]
  2. sinxcos3xdx over [π3,π3]
  3. 3x22x3+1dx over [0,1]
  4. Is the substitution u=1x2 in the definite integral 20x1x2dx okay? If not, why not?

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

  1. π0tcos(t2)sin(t2)dt
  2. 1012t(1+(t12)2)dt
  3. 20(1t)cos(πt)dt
  4. Show that the average value of f(x) over an interval [a,b] is the same as the average value of f(cx) over the interval [ac,bc] for c>0.
  5. Find the area under the graph of g(t)=t(1t2)a between t=0 and t=x, where [latex]00[/latex] is fixed. Evaluate the limit as x1.
  6. The area of the top half of an ellipse with a major axis that is the x-axis from x=1 to a and with a minor axis that is the y-axis from y=b to b can be written as aab1x2a2dx. Use the substitution x=acost to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.
  7. The following graph is of a function of the form f(x)=acos(nt)+bcos(mt). Estimate the coefficients a and b and the frequency parameters n and m. Use these estimates to approximate π0f(t)dt.

    The graph of a function of the given form over [0, 2pi]. It begins at (0,1) and ends at (2pi, 1). It has five turning points, located just after pi/4, between pi/2 and 3pi/4, pi, between 5pi/4 and 3pi/2, and just before 7pi/4 at about -1.5, 2.5, -3, 2.5, and -1. It crosses the x axis between 0 and pi/4, just before pi/2, just after 3pi/4, just before 5pi/4, just after 3pi/2, and between 7pi/4 and 2pi.

Integrals Involving Exponential and Logarithmic Functions

In the following exercises, verify by differentiation that lnxdx=x(lnx1)+C, then use appropriate changes of variables to compute the integral.

  1. x2ln2xdx
  2. lnxxdx(Hint:  Set u=x.)
  3. Write an integral to express the area under the graph of y=et between t=0 and t=lnx, and evaluate the integral.

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

  1. sin(3x)cos(3x)sin(3x)+cos(3x)dx
  2. xcsc(x2)dx
  3. ln(cscx)cotxdx

In the following exercise, evaluate the definite integral.

  1. 211+2x+x23x+3x2+x3dx

In the following exercises, integrate using the indicated substitution.

  1. y1y+1dy;u=y+1
  2. sinx+cosxsinxcosxdx;u=sinxcosx
  3. ln(x)1(lnx)2xdx;u=lnx

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area.

  1. y=ex over [0,1]
  2. y=x+1x2+2x+6 over [0,1]
  3. y=2x over [0,1]

Integrals Resulting in Inverse Trigonometric

In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

  1. 3/20dx1x2

  2. 13dx1+x2
  3. 21dx|x|x21