13. [0/1 Points] DETAILS PREVIOUS ANSWERS TANAPCALC10 6.5.038. Find the average value of the function f over the interval [-1, 9]. f(x) = 1 - x2 -23.34 X Need Help? Read It Watch It\f2. [3/6 Points] DETAILS PREVIOUS ANSWERS TANAPCALC10 6.2.002.MI.SA. This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Find the indefinite integral. 8 x ( 4 * 2 + 8 ) 2 dx Step 1 In this situation, finding the indefinite integral is most easily achieved using the method of integration by substitution. The first step in this method is to let u = g(x), where g(x) is part of the integrand and is usually the "inside function" of a composite function f(g(x)). For the given indefinite integral / 8x(4x2 + 8)2 dx, observe that the integrand involves the composite function (4x2 + 8)2 with the "inside function" g(x) = 4x2 + 8. Therefore, we will choose u = 4x2 + 8 8 Step 2 The next step is to find du = g'(x) dx. First find g'(x). g(x) = 4x2 + 8 g'(x) = 8x 8.r Therefore, we have the following. du = 8.x dx 8x Step 3 We now use the substitution u = 4x2 + 8 and du = 8x dx to convert the entire integral into one involving only u. /8 x ( 4x 2 + 8 ) 2 dx - / ( 4x 2 + 8 ) 2 (8 x dx ) = / du Submit Skip (you cannot come back)4. [l6 Points] TANAPCALC10 6.2.006.MI.SA. This question has several parts that must be Completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part, Tutorial Exercise Find the indefinite integral. / 4x3 + a d 7 X (x4 + Bx)2 mm Need Help? _ 6. [1/6 Paints] PREVIOUS ANSWERS TANAPCALC10 6.4.016.M|.SA. This question has several parts that must be completed sequential/y Ifyou skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skr'pped part. Find the area of the region under the graph of the function fan the intervai [8, 9], x) : Sex 7 x Step 1 Recall the Fundamental Theorem of Calculus, which states that if f is continuous on [a, b], then the foilowmg holds, where F is any antiderivative of f. That is, F'(X) : x). I! / f(x) dx : F(b) 7 F(a) a The Fundamental Theorem of Caiculus can be used to find the area ofthe region under the graph of the function f(x) 7 Sex 7 x on the interval [a, b] 7 [8, 9]. The area of the region is given by the following 9 A: x7 :1 (5e x 511i_L)X To evaluate the definite integral, we first find an antiderivative Fof f(x] = Sex 7 x. Doing so gives the foilowing resuit, where C is an arbitrary constant. 2 F(x):5ex7( L )+c X x In the next step, we wiil evaluate the definite integral. Recail that the constant of integration will "drop out\" when this computation is performed, so we may drop the Cat this point. Doing so gives the following result, Rx) : 5e)r 7( ) Submit suyou cannot come back) 2 2 We have determined that f4xex dX : 2eX + C. Recall that the constant of integration will "drop out" when the definite integral is evaluated, 50 we may drop the c at this point. Using this result, we now evaluate the given definite integral. 3 2 z 2 7 x / 4m" dx ' 25 H o u Submit lyou cannot come back) ~eeu Help