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Integrales interesantes
1.
EJERCICIO NO. 1.
APLICACIΓN DE FΓRMULAS BΓSICAS Y TΓCNICAS ALGEBRAICAS. INSTRUCCIONES: DETERMINA EL RESULTADO DE LAS SIGUIENTES INTEGRALES Y COMPRUEBA LOS RESULTADOS MEDIANTE LA DIFERENCIACIΓN. β« π π π β« ππ π π β« ππ π π β« βπ + π π π β« βπ π π β« π βπ β π π π β« ππ π β π π π β« π₯π§ ππ πΊππ ππ π π β« πππ π π β« πππ π π β« βπππ π π β« πππ π π β« βπ π π β« βπβπ π π β« ππ π π π π β« ππ π π π π β« ββππ π π β« πβππ π π β« βπππ π π π π β« βππβππ π π π β« ππ π π π β« βπ π π π β« βππ π π π β« βπππ πβπ π π β« πβπ π π β« πβπ π π β« π π ππ β« π π ππ β« ( π π β π ππ ) π π β«(ππ β π) π π β«(πππ β ππ + π) π π β« ( π βπ π β πβπ π ) π π β« ( π ππ β π ππ + π π ) π π β« ( π π π π β π π π π + π π π π ) π π β«(πππ β πππ + π) π π β« (π π π β ππ π π + π) π π β«(ππ β ππ β π) π π β« ( ππ π β ππ π ) π π β« ( ππ β πππ β π π ) π π β« ( πππ + ππ + π ππ ) π π
2.
β« ( π βππ π β π βπ π )
π π β« ( π π π β π π π β π π π π π π ) π π β« ( πβ ππ π β ππ π π β π πβπ ) π π β«(ππ β π)π π π β«(ππ β π)π π π β«(ππ β π)π βπ π π β«(ππ + π)π (β π π ) π π β« ππ π (ππ β πβπ) π π π β« ππ π (ππ β ππ)π π π β« (ππ β π ππ ) π π π β« (βπ β π βπ ) π π π β«(βπ β βπ) π π π β«(ππ β π)π (β ππ π π ) π π β« (π + π ππ ) π ( π ππ ) π π β« βππ (β π π ) π π π β« πβπ + πβπ + πβπ πβπ π π β« (ππ + π)π βπ π π π β« π(ππ + π)π βππ π π π β«(πππ β πππ β ππ + π) π π β«(πππ β πππ + π β π) π π β«(πππ β πππ + πππ β π + π) π π β«(πππ β πππ + πππ β ππ + π) π π β« (ππ π π β ππ π π + ππ π π β ππ + π) π π β« (ππ π π β ππ π π + ππ π π β π π π) π π β« ( ππ βππ + ππ β ππ βπ + πβπ ππ β π π + π) π π β« ( ππ πππ ππ + πππ π₯π§ π β π ππ ) π π EJERCICIO NO. 2 APLICACIΓN DEL MΓTODO DEL CAMBIO DE VARIABLE. INSTRUCCIONES: DETERMINA EL RESULTADO DE LAS SIGUIENTES INTEGRALES Y COMPRUEBA LOS RESULTADOS POR DIFERENCIACIΓN.
3.
β«(ππ + π)ππ π π
β«(ππ β π)π π π β«(πππ β π)ππ π π π β«(ππ + π)π π π π β« βππ + π π π β« βππ β π π π β« βππ β π π π π β« βπππ β π π π π π β« π π π ππ β π β« π π π β π β« π π π (ππ β π)π β« π π βππ β π π β« π π π (πππ β π) β« π π π πππ + π β« π π π (ππ β π)π β« (βπ β π) π βπ π π β« (βπ + π) π π π βπ β« π π βπ(βπ β π) β« ππ π π ππ β ππ β« ππ π π πππ β π β« ππ π π πππ β π β« (ππ β π) (ππ β ππ) π π β« (ππ β πππ) π π (ππ β πππ)π β« ππ β π βππ β ππ + ππ π π β« π π π (π β πππ)π β« π βππ β π π π β« π(π + ππ) π π π π β« π(π β πππ)π π π β« ππ π π βππ β π β« ππ π π βππ + ππ β« π π (π + ππ)π β« π π π (π + πππ)π β« ππ (π + πππ)π π π β« (ππ + π)π π βππ + ππ β« (ππ + π)π π ππ + ππ β«(ππ + π) π πππ π π β« ππ π π (ππ + π) β« ππβπ β πππ π π β« (π + π)π π (ππ + ππ)π β« βπ β ππ π ππ π π β« βπ β ππ π ππ π π β« (π + ππ) π π (ππ + ππ)π β« ππβπβπ + πππ π π β« πππβπβπππβπ + π π π β« ππβπ π π (πππ + ππ)πβπ β« π½πβπ π π½ (π½πβπ + π)ππβπ β« ππ π π (ππ + π)π β« ππβπ π π (ππ + π)πβπ
4.
β« ππβπππ+π +
π π π β«(βπ β π) π π π β«(βπ β π) π π π β«(πβπ β π) π π π β« (βπ π β π) π π π π π π π β« πΊπππ π πͺππ π π π β« πͺππ ππ πΊπππ ππ π π β« πΊππ ππ½ πππ ππ½ π π½ ππΊππ ππ½ β π β« πΊπππ ππ π π βπ + π πππ ππ β« πππ π π πππ + π β« πππ π π πππ + π β« πβππ πππ π π (πππ β π) β« ππππ(ππ)πΊπππ (ππ)π π β« πΊππ (ππ π)πͺπππ(ππ π)π π β« ( πΊππ ππ π + πππ ππ ) π π π β« πΊππ π π π βπͺπππ π π β« πͺππ ππ(π β πΊππ ππ)π π π β« πͺπππ πβπͺππ π + π π π β« πππ π πππ π π π βπππ π β π β« πͺππ ππ π β πΊππ ππ π π β« πΊπππ ππ πͺππ ππ π π β« πΊπππ ππ πͺππ ππ π π β« πͺππ ππ π π βπ + πΊππ ππ β« πΊππ ππ π π πβπβπͺππ ππ + π β« πΊππ ππ π π βπ β πͺππ ππ β« πΊππ π πͺππ π π π (βπͺππππ β π) π β« πΊπππ π πͺππ π π π β« πΊππ π πͺπππβπ π π π βπͺππππ β« πΊπππ ππ πͺππ ππ π π β« πͺπππ π πͺπππ π π π β« πͺπππ π πͺπππ π π π β« πͺππ ππ π π πΊππ ππ + π β« π π πΊπππππβπ β πͺππ ππ β« π πΊππ π πͺππ π π π βπ β π πΊπππ π β« πΊππ ππ π π βπ β πͺπππ π β« πΊπππ π πΊππ ππ π π β« πΊππ ππ π π π β πͺππ ππ β« ππππ ππ π π π + ππ πππ ππ β« πΊππ ππ½ π π½ πͺππ ππ½ β π β« πͺπππ β π β βππͺππ β + π β« πππ(π β ππππ) π π β« (π₯π§ π + π)π π π π β« (π₯π§ ππ + π)π π π π
5.
β« [π β π
π₯π§ (π + π)]π π π π + π β« πππβπππ β π π π β« πβπβπβπ β π βπ π π β« π π π ππ π β« π π π πππ π β« π π π πππ π β« πππ π π πππ + π β« πππ π π ππππ β π β« πππ[ππ(π + πππ)] π + πππ π π β« β π β βπ π ππ π π π β« (π + π π ) π ( π ππ ) π π β« β π + πβπ π π π β« βπ β π π π π ππ β« (ππ β ππ)π π βππ β ππππ + π β« (π + π π ) π π π ππ β« π π βπ(βπ + π) π β« (ππ + π)π π (ππ + ππ) β« (π + π)π π ππ + ππ β« ππ + π π + π π π β« π + π ππ + π π π β« ππ + π π + π π π β« ππ + π π + π π π β« πππ + π π β π π π β« ππ β π ππ + π π π β« ππ π π π + π β« πππ + π πππ β π π π β« π π ππ + π β« π π πππ β π β« ( π π β π + π π β π ) π π β« ( π ππ β π β π ππ β π ) π π β« ( π π ππ β π β π π ππ β π + π π ππ β π + π π π β π ) π π β« π π πβπΊππ ππ(ππΊππ ππ + π)πΊπππ ππ πͺπππ ππ EJERCICIO NO. 3 INTEGRACIΓN DE FUNCIONES EXPONENCIALES. CAMBIO DE VARIABLE EN EL EXPONENTE. INSTRUCCIONES: DETERMINA EL RESULTADO DE LAS SIGUIENTES INTEGRALES Y COMPRUEBA LOS RESULTADOS POR DIFERENCIACIΓN.
6.
β« ππ π π β«
πππ π π β« πβπ βπ π π β« πππ π π π β« ππΊππ π πͺππ π π π β« π π π π π β« πππ ππ π π π β« βπππ π π π β« π π πππ β« π π πβππ β« πππ πππ π π β« {β[βπππ π ]} π π π β« ππ πππ π π β« ππ πππ π π β« πππ πππ π π β« (πβ π π + π π π) π π β« π π πΊππππ πΊππ ππ π π β«(πππ β π)π π π β« ππππ ππ πΊπππ ππ β« ππ πππ π π β« ππ (π β πππ ) π π β« ππππ πΊππ ππ βπ β πππ π π β« ( ππππ πππ πππ βπ β ππππ ) ππ π π β« ππΊππ ππ π π πΊππ ππ β« ππππ ππ π π πͺπππ ππ β« ππππ πππ(πͺππ(πππππ) πͺπππ ππ πΊπππππ π π β« ( πππ β πππ ππππ ) π π β« ππͺππππ π π πͺππ ππ β« πππβπΊππππ(π β πͺππππ)π π β« πππ βπππΊππ π ( πΊππ π πͺππ π + πΊππ π ) βπ π π β« ππππ πππ§( π πβπ ) ππ β π β ππ π π β« ππ+ππ π π β«(ππβππ β πππ)π π π EJERCICIO NO. 4 INTEGRACIΓN DE FUNCIONES TRIGONOMΓTRICAS. INSTRUCCIONES: DETERMINA EL RESULTADO DE LAS SIGUIENTES INTEGRALES Y COMPRUEBA LOS RESULTADOS POR DIFERENCIACIΓN.
7.
β« πͺππ ππ
π π β« πππ ππ π π β« πͺππ ππ π π β« πͺππ ππ π π β« πΊππ ππ πππ ππ π π β« πͺππ ππ πͺππ ππ π π β« πΊπππ ππ π π β« πͺπππ ππ π π β« ππππ ππ π π β« π π π πΊπππ(ππ) β« π π βπ(πͺπππ[βπ]) β« πΊππ (π) π π πͺπππ(π) β« π πΊππ(πππ)π π β« ππ πͺππ(ππ) π π β« πππ ππππ (ππ β π) π π β« π π [πΊππ(ππ π)] π π β« ππ(πͺππ[ππ]) π π β« π ππ + π πΊππ(πππ πππ π) π π β«(πππ ππ + πππ ππ)π π π β« (πͺππ ππ(ππ) β πͺππ ππ(ππ)) π π π π β«[πππ(ππ) β πππ(ππ)]π ππ π π β« πΊπππ π π π π β πͺππ π β« πͺπππ ( π π ) πΊππ ( π π ) π π β« βπ + πͺππ π π π β« βπ + πΊππ ππ π π β« π π πͺπππ π β πͺππ ππ β« π + πΊπππ π π + πͺππ ππ π π β«(ππππ π½ + ππππ π½) π π½ β« ππππ π½ βπΊππ π½ π π½ β« π π πΊπππ π(π β π πππ π) β« πππ πΊππ(πππ) π π β« π πππ π π β πππππ π π β« π π πΊππ ( π π ) πππ ( π π ) EJERCICIO NO. 5 INTEGRALES DE FUNCIONES RACIONALES Y COMPLETAR T.C.P. INSTRUCCIONES: DETERMINA EL RESULTADO DE LAS SIGUIENTES INTEGRALES Y COMPRUEBA LOS RESULTADOS POR DIFERENCIACIΓN.
8.
β« π π ππ + ππ β« π
π π ππππ + ππ β« π π ππππ β π β« π π ππππ β ππ β« π π βπ β ππππ β« πΊπππ π π βππ β πͺππππ β« π π βπππ + ππ β« π½π π½ βππ½π + π β« π π π β πππ β« π π π π β ππ β« π π πβππ β π β« π π πβπππ β ππ β« π π βππ β π β« π π βπ β ππ β« βπ β ππ π π β« βππ β πππ π π β« βππ β πππ π π β« βππ + π π π β« βππ + ππ π π β« βππ β ππ π π β« ππ π π βπππ β π β« πͺππ π π π βππ β π πͺππππ β« π π π(πππ π + π) β« π π πβππππ + ππ β« π π (π β π)π + ππ β« π π (π β π)π β π β« π π π β (π β π)π β« π π (π β π)β(π β π)π β π β« β(π β π)π β ππ π π β« π π βπ β (π β π)π β« β(π β π)π β ππ π π β« β(π β π)π + π π π β« π π ππ + ππ + π β« π π ππ β ππ + ππ β« π π ππ + ππ + π β« π π ππ + ππ β« π π ππ β ππ β« π π ππ + ππ + π β« π π ππ + ππ + π β« π π ππ + ππ + ππ β« π π ππ + ππ β π β« π π πππ β ππ + π β« π π πππ + ππ + π β« π π πππ β ππ + π
9.
β« π π πππ + ππ
+ π β« π π πππ + ππ + π β« π π ππ β ππ β ππ β« π π ππππ + πππ + ππ β« π π πππ β ππ β πππ β« π π π + ππ β πππ β« π π βππ β π β ππ β« π π βππ + ππ β ππ β« π π βπππ + ππ β« π π βπππ β π β πππ β« π π βπππ β ππ + π β« π π βπππ β πππ + π β« π π βπππ + πππ + π β« π π βπ β ππ β πππ β« π π βππ β ππ π π β πππ β« βπππ + ππ β π π π β« βπ β ππ β ππ π π β« βππ β ππ β π π π β« βπππ β ππ π π β« πβππ β ππ β ππ π π β« βπ β ππ β πππ π π β« (ππ + π) π π ππ + π β« (ππ β π) π π ππ β π β« (ππ + π) π π βππ β ππ β« (ππ β π) π π π β ππ β« (π β ππ)π π πππ β ππ β« (ππ β π)π π πππ β π β« (ππ β π) π π ππ + π β« (π + π) π π βππ + π β« (ππ β π)π π ππ β ππ β« (π + π)π π ππ β ππ β« (ππ β π)π π πππ + ππ + π β« (π + π) π π ππ β ππ + π
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