Answer to Question #223111 in Calculus for Nama Glory

Question #223111

Evaluate the following Integrals

3)  ʃ-11 (x + 1/2) √(x2+x+1)dx

4)  ʃ (x+2) / √(x+3) dx



1
Expert's answer
2021-09-14T06:05:55-0400


3)


11(x+12)(x2+x+1)dx\int_{-1}^1 (x +\frac{1}{2}) \sqrt{(x^2+x+1)}dx


Solution

(x+12)(x2+x+1)dx\int (x +\frac{1}{2}) \sqrt{(x^2+x+1)}dx


Substitute. u=x2+x+1u=x^2+x+1

dudx=2x+1dx=12x+1du:\frac{du}{dx}=2x+1 \to dx=\frac{1}{2x+1}du:


=12udu=\frac{1}{2} \int \sqrt{u}du


Now solving:


=udu=\int \sqrt{u}du


Apply power rule


=undu=un+1n+1=\int u^ndu=\frac{u^{n+1}}{n+1} with n=12n=\frac{1}{2}


=2u323=\frac{2u^{\frac{3}{2}}}{3}


Plug in solved integrals


=12udu=\frac{1}{2} \int \sqrt{u}du


=u323=\frac{u^{\frac{3}{2}}}{3}


Undo substitution u=x2+x+1:u=x^2+x+1:


=(x2+x+1)323+C=\frac{(x^2+x+1)^{\frac{3}{2}}}{3}+C


Inserting the bounds

=[(12+1+1)323+C][((1)2+(1)+1)323+C]=[\frac{(1^2+1+1)^{\frac{3}{2}}}{3}+C] -[\frac{((-1)^2+(-1)+1)^{\frac{3}{2}}}{3}+C]


313=1.398717474235544\sqrt{3}-\frac{1}{3}=1.398717474235544






4)

(x+2)(x+3)dx\smallint \frac{(x+2) }{\sqrt{(x+3)}} dx


Solution

x+2x+3dx\smallint \frac{x+2}{\sqrt{x+3}} dx

Substitute u=x+3u=x+3

dudx=1dx=du\frac{du}{dx}=1 \to dx=du


=u1udu=\smallint\frac{u-1}{\sqrt{u}}du


Expand


=[u1u]du=\smallint[{\sqrt{u}}-\frac{1}{\sqrt{u}}]du


Apply linearity


=udu1udu=\smallint{\sqrt{u}}du - \smallint \frac{1}{\sqrt{u}}du


Now solving


=udu=\smallint \sqrt{u} du


Apply power rule:


=undu=un+1n+1=\int u^ndu=\frac{u^{n+1}}{n+1} with n=12n=\frac{1}{2}


=2u323=\frac{2u^{\frac{3}{2}}}{3}


Now solving


1udu\int\frac{1}{\sqrt{u}}du


Apply power rule with n=12:n=-\frac{1}{2}:


=2u=2\sqrt{u}


Plug in solved integrals

=udu1udu=\smallint{\sqrt{u}}du - \smallint \frac{1}{\sqrt{u}}du


=2u3232u=\frac{2u^{\frac{3}{2}}}{3}-2\sqrt{u}


Undo substitution u=x+3:u=x+3:


=2(x+3)3232x+3=\frac{2(x+3)^{\frac{3}{2}}}{3}-2\sqrt{x+3}


=2(x+3)3232x+3+C=\frac{2(x+3)^{\frac{3}{2}}}{3}-2\sqrt{x+3} +C



=2[2(x+3)323x+3]+C=2[\frac{2(x+3)^{\frac{3}{2}}}{3}-\sqrt{x+3} ]+C


=2x(x+3)3+C=\frac{2x(x+3)^{}}{3}+C


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment