integraldisplay 2 1 x 1 2 x dx 973 integraldisplay 4 x 2 x 1 dx 974

Integraldisplay 2 1 x 1 2 x dx 973 integraldisplay 4

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integraldisplay 2 1 ( x 1) 2 x dx 973. integraldisplay 4 0 x 2 x + 1 dx 974. integraldisplay π/ 2 0 cos( 2 x 3 ) dx 975. integraldisplay π/ 2 π/ 3 ( x + cos x ) dx 976. integraldisplay 7 0 x 3 x + 1 dx 977. integraldisplay 6 2 x 2 3 x + 2 dx Find the area under the curve over the given interval. 978. y = 2 sin x + sin(2 x ); [0 , π ] 979. y = sin x + cos(2 x ); [0 , π ] 980. y = sec 2 ( x 2 ); [ π 2 , 2 π 3 ] 981. y = csc(2 x ) cot(2 x ); [ π 12 , π 4 ] No one really understood music unless he was a scientist, her father had declared, and not just any scientist, either, oh, no, only the real ones, the theoreticians, whose language is mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. “And relationships,” he had told her, “contained the essential meaning of life.” —Pearl S. Buck, The Goddess Abides, Part 1
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110 The AP CALCULUS PROBLEM BOOK 4.9 Integrals Involving Logarithms and Exponentials Find the following indefinite integrals. 982. integraldisplay 1 x + 1 dx 983. integraldisplay x x 2 + 1 dx 984. integraldisplay x 2 4 x dx 985. integraldisplay x 2 + 2 x + 3 x 3 + 3 x 2 + 9 x dx 986. integraldisplay (ln x ) 2 x dx 987. integraldisplay 1 x + 1 dx 988. integraldisplay x x 3 dx 989. integraldisplay 2 x ( x 1) 2 dx 990. integraldisplay cos θ sin θ 991. integraldisplay csc(2 θ ) 992. integraldisplay cos θ 1 + sin θ 993. integraldisplay sec θ tan θ sec θ 1 994. integraldisplay 5 e 5 x dx 995. integraldisplay e x 1 + e x dx 996. integraldisplay e x 1 e x dx 997. integraldisplay e x + e x e x e x dx 998. integraldisplay 5 e x e 2 x dx 999. integraldisplay e sin( πx ) cos( πx ) dx 1000. integraldisplay e x tan( e x ) dx 1001. integraldisplay 3 x dx 1002. integraldisplay 5 x 2 x dx 1003. integraldisplay 3 2 x 1 + 3 2 x dx Find exact values for each of the following definite integrals. 1004. integraldisplay 4 0 5 3 x + 1 dx 1005. integraldisplay 1 1 1 x + 2 dx 1006. integraldisplay e 2 e 1 x ln x dx 1007. integraldisplay 2 0 x 2 2 x + 1 dx 1008. integraldisplay 2 π π 1 cos θ θ sin θ 1009. integraldisplay 5 1 x + 5 x dx 1010. integraldisplay 1 0 e 2 x dx 1011. integraldisplay 3 1 e 3 /x x 2 dx 1012. integraldisplay 2 1 2 x dx 1013. integraldisplay 1 0 3 4 x (4 ln 3) 3 4 x + 1 dx
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CHAPTER 4.INTEGRALS111 4.10 It Wouldn’t Be Called the Fundamental Theorem If It Wasn’t Fundamental In the following four problems, find F ( x ) . 1014. F ( x ) = integraldisplay x 1 1 t dt 1015. F ( x ) = integraldisplay x 0 tan t dt 1016. F ( x ) = integraldisplay 3 x x 1 t dt 1017. F ( x ) = integraldisplay x 2 1 1 t dt 1018. Let f be a continuous function with an antiderivative F on the interval [ a, b ]. Let c be any point in the interval. State whether the following are true or false. If false, then correct the statement or give an example to show why it is false. a)b)c)d)e) integraldisplay b a f ( x ) dx = integraldisplay c a f ( x ) dx + integraldisplay b c f ( x ) dx integraldisplay b a F ( x ) dx = f ( b ) f ( a ) integraldisplay b a f ( x ) dx 0 integraldisplay b a cf ( x ) dx = c ( F ( b ) F ( a )) integraldisplay b a f ( x ) dx = f ( m )( b a ) for some m in [ a, b ] 1019.
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