2cos2 u 1 2 1 cos 4 u du 3 8 u 1 4 sin 2 u 1 32 cos 4

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2cos2u+12(1+cos 4u))du¿38u14sin 2u+132cos 4u+c¿38(x3)14sin2(x3)+132cos 4(x3)+c10.Evaluate the integral.4cotcscxx dxu=cotx ,du=−cs c2xdx¿cotx(1+cot2x)csc2xdx¿(u+u3)du¿u22u44+c¿cot2x2cot44+c18.Evaluate the integral.22(cossin)xx dx1dx=x+c28.Evaluate the integral.321xxdx¿12x2x21(2x)dxu=x21,du=2xdxWA 2, p. 5
¿12(u+1)udu¿12u32+u12du¿12(2u525+2u323)+c¿15(x21)52+13(x21)32+c30.Evaluate the integral.24xdxxx=2sec θ,dx=2sec θtanθdθ¿4sec2θtanθ2tanθ¿2sec2θdθ¿2tanθ+c¿x24+cSection 6.42.Find the partial fractions decomposition and an antiderivative.2524xx¿5x2(x+2)(x2)¿Ax+2+Bx2WA 2, p. 6
5x2=A(x2)+B(x+2)x=−2:12=−4A ; A=3x=2,8=4B;B=25x2x24=3x+2+2x23x+2(¿+2x2)dx5x2x24dx=¿¿2ln|x+2|+2ln|x2|+cWA 2, p. 7
12. Find the partial fractions decomposition and an antiderivative.314xx¿AX+Bx+Cx2+41=A(x2+4)+(Bx+C)x1=A x2+4A+B x2+Cx=(A+B)x2+4A+Cxx=0:A=14,B=14,C=01x3+4xdx¿[14x+14xx2+4]dx¿ln|x|4ln(x2+4)816. Find the partial fractions decomposition and an antiderivative.2269xxx¿2x(x3)2¿Ax3+B(x3)22x=A(x3)+BA=2; B=62xx26x+9=2x3+6(x3)2WA 2, p. 8
(2x3+6(x3)2)dx¿2ln|x3|6x3+c22. Evaluate the integral.

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