Substitution
- Why is uu-substitution referred to as change of variable?
- If f=g∘h, when reversing the chain rule, ddx(g∘h)(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.
- For x>1: ∫x2√x−1dx=215√x−1(3x2+4x+8)+C;u=x−1
- ∫x√4x2+9dx=14√4x2+9+C;u=4x2+9
- ∫(x+1)4dx;u=x+1
- ∫(2x−3)−7dx;u=2x−3
- ∫x√x2+1dx;u=x2+1
- ∫(x−1)(x2−2x)3dx;u=x2−2x
- ∫cos3θdθ;u=sinθ
In the following exercises, use a suitable change of variables to determine the indefinite integral.
- ∫x(1−x)99dx
- ∫(11x−7)−3dx
- ∫cos3θsinθdθ
- ∫cos2(πt)sin(πt)dt
- ∫tsin(t2)cos(t2)dt
- ∫x2(x3−3)2dx
- ∫y5(1−y3)3/2dy
- ∫(1−cos3θ)10cos2θsinθdθ
- ∫(sin2θ−2sinθ)(sin3θ−3sin2θ)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.
- y=x(1−x2)3 over [−1,2]
- y=x(x2+1)2 over [−1,1]
In the following exercises, use a change of variables to evaluate the definite integral.
- ∫10x√1+x2dx
- ∫10t2√1+t3dt
- ∫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.
- ∫cos(ln(2x))xdx on [0,2]
- ∫sinxcos3xdx over [−π3,π3]
- ∫3x2√2x3+1dx over [0,1]
- Is the substitution u=1−x2 in the definite integral ∫20x1−x2dx okay? If not, why not?
In the following exercises, use a change of variables to show that each definite integral is equal to zero.
- ∫√π0tcos(t2)sin(t2)dt
- ∫101−2t(1+(t−12)2)dt
- ∫20(1−t)cos(πt)dt
- 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.
- Find the area under the graph of g(t)=t(1−t2)a between t=0 and t=x, where [latex]0
0[/latex] is fixed. Evaluate the limit as x→1. - 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 ∫a−ab√1−x2a2dx. 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.
- 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.
Integrals Involving Exponential and Logarithmic Functions
In the following exercises, verify by differentiation that ∫lnxdx=x(lnx−1)+C, then use appropriate changes of variables to compute the integral.
- ∫x2ln2xdx
- ∫lnx√xdx(Hint: Set u=√x.)
- 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.
- ∫sin(3x)−cos(3x)sin(3x)+cos(3x)dx
- ∫xcsc(x2)dx
- ∫ln(cscx)cotxdx
In the following exercise, evaluate the definite integral.
- ∫211+2x+x23x+3x2+x3dx
In the following exercises, integrate using the indicated substitution.
- ∫y−1y+1dy;u=y+1
- ∫sinx+cosxsinx−cosxdx;u=sinx−cosx
- ∫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.
- y=e−x over [0,1]
- y=x+1x2+2x+6 over [0,1]
- y=−2−x over [0,1]
Integrals Resulting in Inverse Trigonometric
In the following exercises, evaluate each integral in terms of an inverse trigonometric function.
- ∫√3/20dx√1−x2
- ∫1√3dx√1+x2
- ∫√21dx|x|√x2−1