91Ó°ÊÓ

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$∫\frac{1}{(1+x)(1-x)}dx.$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises $19–64$. Use

a graph to check your answer.

${∫}_0^{5\mathrm{π}/4}\left|\sin x\right|dx$

For each function,f and interval[a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.

$f\left(x\right)={\left(e^x\right)}^2,[a,b]=[−1,1]$

Prove that if $f$has an antiderivative (say, $G$), then the functionrole="math" localid="1648733712891" $A\left(x\right)={∫}_0^{x}f\left(t\right)dt$ must also be an antiderivative of $f$. (Hint: Use the Fundamental Theorem of Calculus.)

Solve each of the integrals in Exercises 63–68, where a, b, and c are real numbers with

$a≠0,b≠0,c>1,andc≠0.$

$∫a{e}^{bx+c}dx.$

Suppose $f$is continuous on all of $R$. Prove that for all real numbers $a$and $b$, the functionsrole="math" localid="1648735230980" $A\left(x\right)={∫}_a^{x}f\left(t\right)dt$ androle="math" localid="1648735266092" $B\left(x\right)={∫}_b^{x}f\left(t\right)dt$ differ by a constant. Interpret this constant graphically.

Solve each of the integrals in Exercises 63–68, where a, b, and c are real numbers with $a≠0,b≠0,c>1,andc≠0.$

$∫\frac{a}{b{x}^c}dx.$

For each function f and interval[a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.

$f\left(x\right)=\frac{1}{3x+1},\left[a,b\right]=\left[2,5\right]$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises $19–64$. Use

a graph to check your answer.

${∫}_{-1}^1\left|e^{2x}-1\right|dx$

The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case $h→0^{+}$. Rewrite this proof in your own words, and then write a proof of what happens as $h→0^{-}$.