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Evaluate the integral.$\int_{0}^{2} x^{3} \sqrt{x^{2}+4} d x$
$\frac{64 \sqrt{2}}{15}+\frac{64}{15}$
Calculus 2 / BC
Chapter 6
TECHNIQUES OF INTEGRATION
Section 2
Trigonometric Integrals and Substitutions
Integration Techniques
Improper Integrals
Missouri State University
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Lectures
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here in this problem, we have evaluated the interval zero to X cube on little X Square plus one, not in the eggs. Now here I'm substituting X us to danger Tito. So you get Deeks as to seeking square Tito Digital. And when X is zero, we get Tita as zero. And when X is too, we get Tita us then inwards. One riches by wherefore so the integral becomes integration. Zero to buy by phone on Dhere to come two. Time to cube time Staying in Cuba. Tito So I'm taking toe cube outside So to become eight on under Route X Square means to square tangent square teeter and I can write four as two square times dx dx Harris to seek and score Tedo dot detailed I'm digging toe outside, then trickle so it'll become 16. Now we can write this tome that is under root Two square tangent square T topless to square I'm taking too common so and I'm taking outside the route so we'll get we're left with one plus change in school Deter on one Palestinians 20 days seeking school, Tito on route off seconds Credit is sick and leader so we'll get two dames secret. Tito, An intruder becomes zero to buy by four Danger Cube Teeter. This is two seconds later. I'm taking you outside the integral. So it is. I'm 32 on Dhere. It is sick and Tita times seeking Squared Peter Norton Meat eater. No, I'm writing this in Dragila's started to add as it is zero to buy. Therefore ancient square Tito times seeking sweet eater and one agent TDM ready here on second theater as it is dark day, Tita. Now we can use on formula the pain it square. Tita is equal to Seacon Square Tinker minus one. So the angel becomes zero to pi by two Seek and square t down minus one times Seacon Square, Tito Times Change and Tita Times seeking Tito dot De Vito Now against substitute you as seeking Tito So you get the U. S seeking Tito Danger Tita Benedito. And when he dies by 22 0 here it should be by before not by by two. It should be by before it headed Children by before. So when Tito zero we get the U. S. One And when Tita is bye bye full! We get a U S route to? No, the integral becomes integration. We left 30 to her should target. So integration 32 is taking out. I'm doing it outside one to root two hair. It'll come you square minus one times you square on This is Do you? Now we get a very decent trickle Simply so tardy too. Onda, let me right more clearly. Integration. Want to root two headed Come your of our four minus you square dot Do you now The sting will be calls to started to the integration off the year The power four is U to the power Fife minus u to the power three divided by three on the limit is from one Joe root toe. Not simply we have to plug the limits so we'll get will be called to 32. First of all other playing roach to so route due to the power Fife and divide by Fife minus route to cube Divide by three. Now we have to plug a lower limit. So one by five minus one by three on this will be called to 64. Divide by 15 under route to plus one with brackets. So finally we can conclude that the Androgel zero toe X Cube Underwood X Square plus four dot dx is equals to 64. Divide by 15 brackets under Andrew route to bless for one. So this is the answer for the given into the
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