By Christopher Thomas

Updated on Oct 1, 2011

To review these concepts, go to Calculus and Integration by Parts Study Guide.

**Calculus and Integration by Parts Practice Questions**

**Evaluate the following integrals using integration by parts, substitution, or basic integration.**

*x*^{5}ln(*x*)*dx*-
*x*sin(*x*)*dx* -
*x*sin(*x*^{2})*dx* - (
*x*+ 3)cos(*x*)*dx* -
*dx* -
*x*^{2}sin(*x*)*dx* - (
*x*^{2}+ sin(*x*))*dx* -
*x*^{2}*e*^{x3 + 1}*dx* -
*x*^{2}*e*^{x}*dx* - (
*x*^{3}+ 3*x*– 1)ln(*x*)*dx* - (
*x*+ ln(*x*))*dx* - √
*x*– 1*dx* -
*x*√*x*– 1*dx* -
*x**e*^{–x}*dx* - √
*x*ln(*x*)*dx* -
*dx* -
*dx* - (
*x*^{2}– 1)cos(*x*)*dx* -
*dx* - sin(
*x*)√cos(*x*)*dx* - sin(
*x*) · lncos(*x*)*dx* -
*e*^{x}cos(*x*)*dx*

**Solutions**

*x*^{6}ln(*x*) –*x*^{6}+*c*, done by parts with*u*= ln(*x*)- –
*x*cos(*x*) + sin(*x*) +*c*, by parts with*u*=*x* - –cos(
*x*^{2}) +*c*, by the substitution*u*=*x*^{2} - (
*x*+ 3)sin(*x*) + cos(*x*) +*c*, by parts with*u*=*x*+ 3 - (ln(
*x*))^{2}+*c*, by substituting*u*= ln(*x*) - –
*x*^{2}cos(*x*) + 2*x*sin(*x*) + 2cos(*x*) +*c*, using parts twice *x*^{3}– cos(*x*) +*c*, by basic integration*e*^{x÷1}=*c*, by substituting*u*=*x*^{3}+ 1*x*^{2}*e*^{x}– 2*x**e*^{x}+ 2*e*^{x}+ c, using parts twice- (
*x*^{4}+*x*^{2}–*x*)ln(*x*) –*x*^{4}–*x*^{2}+*x*+*c*, by parts with*u*= ln(*x*) *x*^{2}+*x*ln(*x*) –*x*+*c*, evaluating, ∫ ln(*x*)*dx*by parts- +
*c*, substituting*u*=*x*– 1 - +
*c*, by parts with*u*=*x* - –
*xe*^{–x}–*e*^{–x}+*c*, by parts with*u*=*x* *x*^{}ln(*x*) –*x*^{}+*c*, by parts with*u*=*x*- – – +
*c*, by parts with*u*= ln(*x*) - lnl
*x*l +*c*, by basic integration - (
*x*^{2}– l)sin(*x*) + 2*x*cos(*x*) – 2sin(*x*) +*c*, using parts twice - +
*c*, by substituting*u*= - , by substituting
*u*= cos(*x*) - –cos(
*x*) · ln(cos(*x*)) + cos(*x*) +*c*, by parts with*u*= ln( cos(*x*)) . This could also be solved by substitution with*u*= cos(*x*) , though it would require knowing ln(*u*)*du*. - (
*e*^{x}sin(*x*) +*e*^{x}cos(*x*)) +*c*, by parts twice, plus the trick from the previous example

From Calculus Success in 20 Minutes A Day. Copyright © 2006 by LearningExpress, LLC. All Rights Reserved.

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