有理函数的不定积分习题.doc

SOLUTION 1 : Integrate . First, split this rational function into two parts. Thus, (Now use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 2 : Integrate . Use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , getting (Now use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 3 : Integrate . Rewrite the function and use formula 3 from the introduction to this section. Then . Click HERE to return to the list of problems. SOLUTION 4 : Integrate . Use u-substitution. Let so that , or . Substitute into the original problem, replacing all forms of , getting (Now use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 5 : Integrate . First, use polynomial division to divide by . The result is . In the second integral, use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , getting (Now use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 6 : Integrate . First, use polynomial division to divide by . The result is . In the third integral, use u-substitution. Let so that , or . For the second integral, use formula 2 from the introduction to this section. In the third integral substitute into the original problem, replacing all forms of , getting (Now use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 7 : Integrate . Use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , getting (Use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 8 : Integrate . Use u-substitution. Let so that . In addition, we can back substitute with . Substitute into the original problem, replacing all forms of , getting (Combine and since is an arbitrary constant.) . SOLUTION 9 : Integrate . First, complete the square in the denominator. The result is . Now use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , getting (Use formula 2 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 10 : Integrate . First, factor 2 from the denominator. The result is (Complete the square in the denominator.) . Use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , getting (Use formula 3 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 11 : Integrate . Because of the term in the denominator, rewrite the term in a somewhat unusual way. The result is . Now use u-substitution. Let so that , or . Substitute into the original problem, replacing all forms of , getting (Use formula 3 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 12 : Integrate . Use u-substitution. Let so that (Dont forget to use the chain rule on .) , or . Substitute into the original problem, replacing all forms of , and getting (Use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 13 : Integrate . First, rewrite the denominator of the function, getting . Now use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , and getting (Use formula 2 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 14 : Integrate . Use u-substitution. Let so that (Dont forget to use the chain rule on .) , or . Substitute into the original problem, replacing all forms of , and getting (Use formula 1 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 15 : Integrate . First, rewrite the denominator of the function, getting (Recall that .) . Now use u-substitution. Let so that . Substitute into the original problem, replacing all forms of , and getting (Use formula 2 from the introduction to this section.) . Click HERE to return to the list of problems. SOLUTION 16 : Integrate . Use u-substitution. Let so that , or . In addition, we can back substitute with . Substitute into the original problem, replacing all forms of , getting (Combine and since is an arbitrary constant.) . SOLUTION 17 : Integrate . First factor the denominator, getting . Now use u-substitution. Let so that . In addition, we can back substitute with . Substitute into the original problem, replacing all forms of , getting . Click HERE to return to the list of problems. SOLUTION 18 : Integrate . First complete the square in the denominator, getting . Now use u-substitution. Let so that . In addition, we can back substitute with . Substitute into the original problem, replacing all forms of , getting . In the first integral use substitution. Let so that , or . Substitute into the first integral, replacing all forms of , and use formula 3 from the beginning of this section on the second integral, getting . Click HERE to return to the list of problems. SOLUTION 19 : Integrate . First factor out a 2 and complete the square in the denominator, getting . Now use u-substitution. Let so that . In addition, we can back substitute with . Substitute into the original problem, replacing all forms of , getting . In the first integral use substitution. Let so that , or . Substitute into the first integral, replacing all forms of , and use formula 3 from the beginning of this section on the second integral, getting . Click HERE to return to the list of problems. SOLUTION 20 : Integrate . First rewrite this rational function by multiplying by , getting (Recall that