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Ejercicios

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The document provides examples of using integration by parts to calculate integrals of various functions. It gives the steps to calculate integrals of functions such as ye^2y dy, seny dy, cos^3y dy, lny dy, arctany dy, and others. For each integral it identifies the u and dv factors, performs the integration by parts calculation, and arrives at the solution.

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Ejercicios<br />Utilice la integraci¨®n por partes para calcular las integrales de los siguientes problemas.<br />ye2ydy; Sea u=y ; du=dy ; dv=e2ydy ; v=12e2y<br />ye2ydy=12e2-12e2ydy=12ye2-14e2y+c<br />y seny dy; Sea u=y; du=dy; dv=seny; v=-cosy<br />y seny dy=-y cosy+cosy dy=-y cosy+seny+c<br />y cos3y dy; Sea u=y; du=dy; cos3y dy; 13cos3y<br />y cos3y dy=13y seny-13sen3y dy=13y seny- 19cos3y+c<br />y3lny dy; u=lny; du=dyy; dv=y3; v=y44<br />y3lny dy=14lny-14y3dy=14lny-116y4+c<br />arctany dy; Sea u=arctany; du=dyy2+1; dv=dy; v=y<br />arctany dy=y arctany-y dyy2+1; Sea s=1+y2; ds2=y dy<br />arctany dy=y arctany-12dss=y arctany-12lns+c<br />arctany dy=y arctany-12lny2+1+c<br />y lny dy; Sea u=lny; du=dyy; dv=y; v=23y32<br />y lny dy=23y32lny-23ydy=23y32lny-49y32+c<br />ln2t dt; Sea u=ln2t; du=2lnt dtt; dv=dt; t=v<br />ln2t dt=t lnt-2lnt dt; Sea r=lnt; dr=dtt; ds=dt; s=t<br />ln2t dt=t ln2t-2t lnt-dt=t ln2t-2t lnt+2t+c<br />t t+3 dt; Sea u=t; du=dt; dv=t+3 dt; v=23t+332<br />t t+3 dt=23tt+332-23t+332dt=23tt+332-415t+352+c<br />x5x3+1dx; Sea u=x3+1; x=u2-11/3; 23du=x2dxx3+1 du=3x2dx2x3+1; dv=x5dx; v=x66<br />x5x3+1dx=16x6x3+1-14x8dxx3+1=16x6x3+1-16u2-12 du=<br />16x6x3+1-16u4-2u2+1du=16x6x3+1-130u52+19u-u12+c<br />x5x3+1dx=16x6x3+1-130x3+15/2+19x3+1-x3+11/2+c<br />csc3¦È d¦È=csc2¦È csc¦Èd¦È; Sea u=csc¦È; du=-csc¦È cot¦È d¦È; dv=csc2¦È d¦È; v=-cot¦È<br />csc3¦È d¦È=-csc¦È cot¦È-csc¦È cot2¦Èd¦È; Como csc2¦È-1=cot2¦È entonces:<br />csc3¦È d¦È=-csc¦È cot¦È-csc3¦È d¦È+csc¦Èd¦È; Despejamos csc3¦È d¦È, entonces:<br />2csc3¦È d¦È=-csc¦È cot¦È+csc¦Èd¦È; Ahora analicemos la csc¦Èd¦È.<br />csc¦Èd¦È=csc¦Ècsc¦È+cot¦Ècsc¦È+cot¦Èd¦È; Sea u=csc¦È+cot¦È; -du=csc¦Ècsc¦È+cot¦Èd¦È<br />csc¦Ècsc¦È+cot¦Ècsc¦È+cot¦Èd¦È=-duu=-lncsc¦È+cot¦È+c; Por lo tanto:<br />2csc3¦È d¦È=-csc¦È cot¦È-lncsc¦È+cot¦È<br />csc3¦È d¦È=12-csc¦È cot¦È-lncsc¦È+cot¦È+c<br />x3arctanx dx; Sea u=arctanx; du=dxx2+1; dv=x2dx; v=x33<br />x3arctanx dx=13x3arctanx-13x3x2+1dx; Sea t-1=x2; dt2=xdx<br />x3arctanx dx=13x3arctanx-16t-1tdt<br />x3arctanx dx=13x3arctanx-16dt+16dtt<br />x3arctanx dx=13x3arctanx-16t+16lnt+c<br />x3arctanx dx=13x3arctanx-16x2+1+16lnx2+1+c<br />arcsect dt; Sea p=t; 2p dp=dt<br />arcsect dt=2p arcsecp dp; Sea u=arcsecp; du=dppp2-1; dv=p dp; v=p22<br />p arcsecp dp=p24arcsecp-14p dpp2-1; Sea s=p2-1; ds2=p dp<br />p arcsecp dp=p24arcsecp-18dss<br />p arcsecp dp=p24arcsecp-14s+c=p24arcsecp-14p2-1<br />arcsect dt=t4arcsect-14t-1+c<br />arctant dt=??<br />t csc2t dt; Sea u=t; du=dt; dv=csc2t dt; v=-cott<br />t csc2t dt=-t cott+cott dt=-t cott+costsentdt; Sea s=sent; du=cost dt<br />t csc2t dt=-t cott+dss=-t cott+lns+c<br />t csc2t dt=-t cott+lnsent+c<br />t3cost2dt; u=t2; du2=t dt; dv=t cost2dt; v=12sent2<br />t3cost2dt=12t2sent2-t sent2dt=12t2sent2-12senu du<br />t3cost2dt=12t2sent2+12cosu+c=12t2sent2+12cost2+c<br />lntttdt; Sea u=lnt; du=dtt; dv=dttt; v=-2t<br />lntttdt=-2 lntt+2dtt=-2 lntt+4t+c<br />t cosht dt;u=t; du=dt; dv=cosht dt; v=-1hsenht<br />t cosht dt=1ht senht-1hsenht dt=1ht senht+1h2cosht+c<br />t2senht dt; Sea u=t2; du=2t dt; dv=senht dt; v=-1hcosht<br />t2senht dt=-2thcosht+2ht cosht dt;Sea r=t; dr=dt; ds=cosht dt; s=1hsenht<br />t2senht dt=-2thcosht+2th2senht-2h2senht dt<br />t2senht dt=-2thcosht+2th2senht+2h3cosht+c<br />
Ejercicios
Ejercicios

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Ejercicios

  • 1. Ejercicios<br />Utilice la integraci¨®n por partes para calcular las integrales de los siguientes problemas.<br />ye2ydy; Sea u=y ; du=dy ; dv=e2ydy ; v=12e2y<br />ye2ydy=12e2-12e2ydy=12ye2-14e2y+c<br />y seny dy; Sea u=y; du=dy; dv=seny; v=-cosy<br />y seny dy=-y cosy+cosy dy=-y cosy+seny+c<br />y cos3y dy; Sea u=y; du=dy; cos3y dy; 13cos3y<br />y cos3y dy=13y seny-13sen3y dy=13y seny- 19cos3y+c<br />y3lny dy; u=lny; du=dyy; dv=y3; v=y44<br />y3lny dy=14lny-14y3dy=14lny-116y4+c<br />arctany dy; Sea u=arctany; du=dyy2+1; dv=dy; v=y<br />arctany dy=y arctany-y dyy2+1; Sea s=1+y2; ds2=y dy<br />arctany dy=y arctany-12dss=y arctany-12lns+c<br />arctany dy=y arctany-12lny2+1+c<br />y lny dy; Sea u=lny; du=dyy; dv=y; v=23y32<br />y lny dy=23y32lny-23ydy=23y32lny-49y32+c<br />ln2t dt; Sea u=ln2t; du=2lnt dtt; dv=dt; t=v<br />ln2t dt=t lnt-2lnt dt; Sea r=lnt; dr=dtt; ds=dt; s=t<br />ln2t dt=t ln2t-2t lnt-dt=t ln2t-2t lnt+2t+c<br />t t+3 dt; Sea u=t; du=dt; dv=t+3 dt; v=23t+332<br />t t+3 dt=23tt+332-23t+332dt=23tt+332-415t+352+c<br />x5x3+1dx; Sea u=x3+1; x=u2-11/3; 23du=x2dxx3+1 du=3x2dx2x3+1; dv=x5dx; v=x66<br />x5x3+1dx=16x6x3+1-14x8dxx3+1=16x6x3+1-16u2-12 du=<br />16x6x3+1-16u4-2u2+1du=16x6x3+1-130u52+19u-u12+c<br />x5x3+1dx=16x6x3+1-130x3+15/2+19x3+1-x3+11/2+c<br />csc3¦È d¦È=csc2¦È csc¦Èd¦È; Sea u=csc¦È; du=-csc¦È cot¦È d¦È; dv=csc2¦È d¦È; v=-cot¦È<br />csc3¦È d¦È=-csc¦È cot¦È-csc¦È cot2¦Èd¦È; Como csc2¦È-1=cot2¦È entonces:<br />csc3¦È d¦È=-csc¦È cot¦È-csc3¦È d¦È+csc¦Èd¦È; Despejamos csc3¦È d¦È, entonces:<br />2csc3¦È d¦È=-csc¦È cot¦È+csc¦Èd¦È; Ahora analicemos la csc¦Èd¦È.<br />csc¦Èd¦È=csc¦Ècsc¦È+cot¦Ècsc¦È+cot¦Èd¦È; Sea u=csc¦È+cot¦È; -du=csc¦Ècsc¦È+cot¦Èd¦È<br />csc¦Ècsc¦È+cot¦Ècsc¦È+cot¦Èd¦È=-duu=-lncsc¦È+cot¦È+c; Por lo tanto:<br />2csc3¦È d¦È=-csc¦È cot¦È-lncsc¦È+cot¦È<br />csc3¦È d¦È=12-csc¦È cot¦È-lncsc¦È+cot¦È+c<br />x3arctanx dx; Sea u=arctanx; du=dxx2+1; dv=x2dx; v=x33<br />x3arctanx dx=13x3arctanx-13x3x2+1dx; Sea t-1=x2; dt2=xdx<br />x3arctanx dx=13x3arctanx-16t-1tdt<br />x3arctanx dx=13x3arctanx-16dt+16dtt<br />x3arctanx dx=13x3arctanx-16t+16lnt+c<br />x3arctanx dx=13x3arctanx-16x2+1+16lnx2+1+c<br />arcsect dt; Sea p=t; 2p dp=dt<br />arcsect dt=2p arcsecp dp; Sea u=arcsecp; du=dppp2-1; dv=p dp; v=p22<br />p arcsecp dp=p24arcsecp-14p dpp2-1; Sea s=p2-1; ds2=p dp<br />p arcsecp dp=p24arcsecp-18dss<br />p arcsecp dp=p24arcsecp-14s+c=p24arcsecp-14p2-1<br />arcsect dt=t4arcsect-14t-1+c<br />arctant dt=??<br />t csc2t dt; Sea u=t; du=dt; dv=csc2t dt; v=-cott<br />t csc2t dt=-t cott+cott dt=-t cott+costsentdt; Sea s=sent; du=cost dt<br />t csc2t dt=-t cott+dss=-t cott+lns+c<br />t csc2t dt=-t cott+lnsent+c<br />t3cost2dt; u=t2; du2=t dt; dv=t cost2dt; v=12sent2<br />t3cost2dt=12t2sent2-t sent2dt=12t2sent2-12senu du<br />t3cost2dt=12t2sent2+12cosu+c=12t2sent2+12cost2+c<br />lntttdt; Sea u=lnt; du=dtt; dv=dttt; v=-2t<br />lntttdt=-2 lntt+2dtt=-2 lntt+4t+c<br />t cosht dt;u=t; du=dt; dv=cosht dt; v=-1hsenht<br />t cosht dt=1ht senht-1hsenht dt=1ht senht+1h2cosht+c<br />t2senht dt; Sea u=t2; du=2t dt; dv=senht dt; v=-1hcosht<br />t2senht dt=-2thcosht+2ht cosht dt;Sea r=t; dr=dt; ds=cosht dt; s=1hsenht<br />t2senht dt=-2thcosht+2th2senht-2h2senht dt<br />t2senht dt=-2thcosht+2th2senht+2h3cosht+c<br />