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MathYou need to evaluate the definite integral using the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_1^2 (1/x^2  4/x^3)dx = int_1^2 1/x^2 dx  int_1^2 4/x^3 dx`...

Math`int_1^4(4+6u)/sqrt(u)du` `=int_1^4(4/sqrt(u)+(6u)/sqrt(u))du` `=int_1^4(4u^(1/2)+6u^(1/2))du` `=[4(u^(1/2+1)/(1/2+1))+6(u^(1/2+1)/(1/2+1))]_1^4` `=[4(u^(1/2)/(1/2))+6(u^(3/2)/(3/2))]_1^4`...

MathYou need to evaluate the definite integral using the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_0^4(3sqrt t  2e^t)dt = int_0^4 3sqrt tdt  int_0^4 2e^t...

MathYou need to evaluate the definite integral, such that: `int_0^1 x(root(3) x + root(4) x)dx = int_0^1 (x^(1+1/3) + x^(1+1/4))dx` `int_0^1 x(root(3) x + root(4) x)dx = ((x^(2+1/3))/(2+1/3) +...

MathYou need to evaluate the indefinite integral, such that: `int f(x)dx = F(x) + c` `int (x^2  x^(2))dx = int (x^2)dx  int x^(2) dx ` Evaluating each definite integral, using the formula `int x^n...

Math`int sqrt(x^3)root(3)(x^2) dx` Before evaluating, convert the radicals to expressions with rational exponents. `= int x^(3/2)*x^(2/3) dx` Then, simplify the integrand. Apply the laws of exponent...

Math`int (x^41/2x^3+1/4x2)dx` To evaluate this integral, apply the formulas `int x^n dx=x^(n+1)/(n+1) +C` and `int adx = ax + C` . `int (x^41/2x^3+1/4x2)dx` `=x^5/5  1/2*x^4/4 + 1/4*x^2/22x...

Math`int (y^3+1.8y^22.4y)dy` To evaluate this integral, apply the formula `int x^n dx = x^(n+1)/(n+1) + C` . `= y^4/4 + 1.8y^3/3  2.4y^2/2 + C` `=0.25y^4 + 0.6y^3  1.2y^2 + C` Therefore, `...

MathYou need to evaluate the indefinite integral, hence, you need to open the brackets, such that: `(u+4)(2u+1) = 2u^2 + 9u + 4` `int (u+4)(2u+1) du = int (2u^2 + 9u + 4) du ` `int (u+4)(2u+1) du =...

Math`intv(v^2+2)^2dv` `=intv((v^2)^2+2v^2*2+2^2)dv` `=intv(v^4+4v^2+4)dv` `=int(v^5+4v^3+4v)dv` apply the power rule, `=v^6/6+4v^4/4+4v^2/2+C` , C is constant `=v^6/6+v^4+2v^2+C`

Math`int(x^32sqrt(x))/xdx` ``Simplify by dividing each term in the numerator by x. ` <br> ` `=(x^3/3)(2x^(1/2))/(1/2)+C` `=(x^3/3)4x^(1/2)+C` `` ` `

Math`int(x^2+1+1/(x^2+1))dx` apply the sum rule, `=intx^2dx+int1dx+int1/(x^2+1)dx` To evaluate the above integrals, we know that, `intx^ndx=x^(n+1)/(n+1)` and `int1/(x^2+1)dx=arctan(x)` using above,...

Math`int(csc^2(t)2e^t)dt` Apply the sum rule, `=intcsc^2(t)dtint(2e^tdt` We now the following common integrals, `intcsc^2(x)=cot(x)` and `inte^xdx=e^x` evaluate using the above, `=cot(t)2e^t+C`...

MathYou need to evaluate the indefinite integral, such that: `int f(theta)d theta = F(theta) + c` `int (theta  csc theta* cot theta)d theta = int theta d theta  int (csc theta* cot theta)d theta`...

MathYou need to evaluate the definite integral using the fundamental theorem of calculus, such that: `int_a^b f(u) du = F(b)  F(a)` `int_0^3 (2sin x  e^x) dx =int_0^3 2sin x dx  int_0^3 e^x dx`...

Math`int_1^2((v^3+3v^6)/v^4)dv` simplify the integrand and apply the sum rule, `=int_1^2(v^3/v^4+(3v^6)/v^4)dv` `=int_1^2(1/v+3v^2)dv` using the following common integrals `int1/xdx=ln(x)` and...

MathEvaluate `int_1^18(3/z)^(1/2)dz` `=int_1^18sqrt(3)z^(1/2)dz` `=sqrt(3)int_1^18z^(1/2)dz` Integrate the function. `=sqrt(3)[z^(1/2)/(1/2)]=sqrt(3)[2z^(1/2)]` Evaluate the function from 1 to...

MathYou need to evaluate the definite integral using the fundamental theorem of calculus, such that: `int_a^b f(x) dx = F(b)  F(a)` `int_(1/(sqrt3))^(sqrt 3) 8/(1+x^2) dx = 8 int_(1/(sqrt3))^(sqrt 3)...

MathYou need to evaluate the definite integral using the fundamental theorem of calculus, such that: `int_a^b f(u) du = F(b)  F(a)` `int_1^2 (4+u^2)/(u^3) du = int_1^2 4/(u^3) du + int_1^2 (u^2)/(u^3)...

Math`int_(1)^1 e^(u+1)du` To evaluate this, apply the formula `int e^x dx = e^x` . `= e^(u+1) _(1)^1` Then, plugin the limits of the integral as follows `F(x) =int_a^b f(x)dx=F(b)F(a)` ....

MathYou need to evaluate the definite integral using the fundamental theorem of calculus such that `int_a^b f(x)dx = F(b)  F(a)` `int_(pi/6)^pi sin theta d theta = cos theta_(pi/6)^pi`...

MathEvaluate `int_5^5(e)dx` Please note that e is a constant approximately equal to 2.718. Integrate the function. `=ex` Evaluate the function from x=5 to x=5. `=e(5)e(5)=5e+5e=10e` =27.183

Math`int_0^1(u+2)(u3)du` `=int_0^1(u^23u+2u6)du` `=int_0^1(u^2u6)du` `=[u^3/3u^2/26u]_0^1` `=[1^3/31^2/26*1][0^3/30^2/26*0]` `=(1/31/26)` `=(2336)/6` =37/6

MathEvaluate `int_0^4(4t)(sqrt(t))dt` `=int_0^4(4t^(1/2)t^(3/2))dt` Integrate the function. `inta^n=a^(n+1)/(n+1)` `=(4t^(3/2))/(3/2)t^(5/2)/(5/2)=(8/3)t^(3/2)(2/5)t^(5/2)` Evaluate the...

Math`int_0^2 (y1)(2y+1)dy` Before evaluating, expand the integrand. `=int_0^2 (2y^2+y2y1)dy` `=int_0^2(2y^2y1)dy` Then, apply the integral formulas `int x^n dx=x^(n+1)/(n+1)` and `int cdx = cx` ....

Math`int_0^(pi/4) sec^2(t) dt` Take note that the derivative of tangent is d/(d theta) tan (theta)= sec^2 (theta). So taking the integral of sec^2(t) result to: `= tan (t) _0^(pi/4)` Plugin the...

Math`int_0^(pi/4) (sec (theta) tan (theta)) d theta` Take note that the derivative of secant is `d/(d theta) (sec (theta)) = sec(theta) tan (theta)` . So taking the integral of sec(theta) tan(theta)...

Math`int_1^2(1+2y)^2dy` `=int_1^2((1)^2+2*2y*1+(2y)^2)dy` `=int_1^2(1+4y+4y^2)dy` `=[y+4y^2/2+4y^3/3]_1^2` `=[y+2y^2+(4y^3)/3]_1^2` `=[2+2(2)^2+(4(2^3))/3][1+2(1)^2+(4(1)^3)/3]`...

Math`int_1^4(52t+3t^2)dt` apply the sum rule and power rule, `=[5t2t^2/2+3t^3/3]_1^4` `=[5tt^2+t^3]_1^4` `=[5*44^2+4^3][5*11^2+1^3]` `=(2016+64)(51+1)` `=(8416)(5)` =63

Math`int_0^4(1+(1/2)u^4+(2/5)u^9)du` `=[u+(1/2)(u^(4+1)/(4+1))+(2/5)(u^(9+1)/(9+1))]_0^4` `=[u+u^5/10u^10/25]_0^4` `[1+1^5/101^10/25][0+0^5/100^10/25]` `=(1+1/101/25)` `=(50+52)/50` =53/50

MathYou need to evaluate the definite integral such that: `int_1^9 sqrt x dx = (x^(3/2))/(3/2)_1^9` `int_1^9 sqrt x dx = (2/3)(9sqrt9  1sqrt1)` `int_1^9 sqrt x dx = (2/3)(271)` `int_1^9 sqrt x dx...

MathEvaluate `int_1^8(x^(2/3))dx` Integrate the function. `inta^n=a^(n+1)/(n+1)` `=x^(1/3)/(1/3)=3x^(1/3)` Evaluate the function from x=1 to x=8. `=3[8^(1/3)1^(1/3)]` `=3[21]` =3

MathYou need to evaluate the integral, such that: `int_(1)^2(x^3  2x)dx = int_(1)^2 x^3 dx  int_(1)^2 2x dx` `int_(1)^2(x^3  2x)dx = (x^4/4  x^2)_(1)^2` `int_(1)^2(x^3  2x)dx = (2^4/4  2*2...

Math`int_(1)^1 x^100 dx` To evaluate this, apply the formula `int x^n dx = x^(n+1)/(n+1)` . `= x^101/101 _(1)^1` Then, plugin the limits of the integral as follows `F(x) =int_a^bf(x) dx = F(b)...

HistoryThere are many different ways to answer this question. Let us look at a few. One way to answer this is to say that Thanksgiving is celebrated as a way of emulating the Pilgrims who were some of...

PoliticsThe problem with defining the term "extremist political organization" is that it is by nature one applied only in the second or third person. In other words, the word "extremism" is one we do not...

Social SciencesCompatibilism attempts to resolve the problem of free will and determinism by proposing that the two positions are compatible. Specifically, it asserts that: Determinism is true, that is, all human...

HistoryLincoln's Thanksgiving Proclamation, which declared the final day of November a national day of "thanksgiving and praise to Almighty God," was one of many similar declarations by American...

HealthTo evaluate an article from the perspective of what you and I find confusing or surprising about the content, we usually have to ask ourselves certain questions: 1)Does the author state his/her...

Of Mice and MenCrooks is the only farmhand who is African American. Living as a black man in the Depression Era, and among a bunch of rogue men of their generation, rendered Crooks as an outsider to the group....

MiseryThe ending of "Misery" is not a happy one, but it is realistic. The poor man can't get anybody to listen to his grief. Nobody cares about him. This seems to be the way of the world. If he were a...

HamletIn Act 1, Scene 1, there is nothing that speaks directly to Hamlet's lack of trust in himself. The scene mostly features the ghost (of Old Hamlet) and how Bernardo, Horatio, and Marcellus react to...

HistoryThis phrase is found in the opening passage of the Declaration, where Jefferson is explaining why a declaration is needed in the first place. In this context "station" means something like "status"...

ScienceThe term "mRNA" stands for "messenger RNA". This is usually enough of a hint to remember their function; to carry messages in the body, specifically messages for creating proteins. The mRNA is like...

MacbethThe queen, Duncan’s wife and Malcolm and Donalbain’s mother, is dead. Although there is a king at the beginning of the play, there is no mention of the queen. Malcolm and Donalbain obviously...

HistoryThe word "imperialism" describes a process where a nation attempts to expand its influence beyond its own borders by gaining control of territories or peoples. This can come in a number of...

Winter DreamsDexter's general feeling about winter is that it engulfs and shadows the beauty of summertime. In the story, we read that Dexter usually breaks out his skis when 'the long, Minnesota winter shut(s)...

On My First SonBefore I answer that question, let's review the definition of "tone". Tone is a literary device that indicates the emotion the author intends to communicate in his/her work, conveyed through word...

A Doll's HouseMain Theme The most salient common theme shared among the female characters found in the short story "The Yellow Wallpaper" by Charlotte Perkins Gilman, the play A Doll's House by Henrik Ibsen, and...

HistoryHow was America's society and political structure different than England's in the late eighteenth...The social structure of the American colonies was far less stratified than that of Great Britain. While there were extremes of wealth, as in the Hudson River Valley, the South Carolina Low Country,...