
Math
Evaluate `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 18....

Math
`int_0^1 (x^e+e^x)dx` To evaluate this, apply the formulas `int u^ndu = u^(n+1)/(n+1)` and `int e^udu=e^u` . `= (x^(e+1)/(e+1) + e^x) _0^1` Then, plugin the limits of integral as follows...

Math
We have to evaluate the inte``gral `\int_{0}^{1}cosh(t)dt` `` We know that the integral of cosh(t) = sinh(t) . Therefore we can write, `\int_{0}^{1}cosh(t)dt=[sinh(t)]_{0}^{1}`...

Math
You 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)...

Math
You 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)` ....

Math
Hello! This integral is a table one, `int((4)/(sqrt(1x^2)))dx=4arcsin(x)+C.` Therefore the definite integral is equal to `4*(arcsin(1/sqrt(2))arcsin(1/2))=4*(pi/4pi/6)=4*pi/12=pi/3 approx 1.047.`

Math
Refer the graph in the attached image. From the graph it appears that area of the region is `~~` 70% of the rectangle. Area of the region=`~~` 70/100(Area of Rectangle) Area of the region...

Math
Refer the graph in the attached image. From the graph, Area of the region that lies beneath the curve `~~` 6/100(Area of Rectangle) Area of the region `~~` 6/100(5*1) `~~` 0.30 Exact Area of the...

Math
Refer the graph in the attached image. From the graph it appears that the area is `~~` 2/3 of the rectangle. Area of the region= `~~` 2/3(Area of rectangle) Area of the region...

Math
You 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`...

Math
Evaluate `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

Math
Evaluate `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 function...

Math
Hello! Find the indefinite integral first: `int((x1)/sqrt(x))dx=int(x^(1/2)x^(1/2))dx=(2/3)*x^(3/2)2*x^(1/2)+C.` So the definite integral is equal to...

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_0^1(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))]_0^1` `=[u+u^5/10u^10/25]_0^1 = [1+1^5/101^10/25][0+0^5/100^10/25]` `=(1+1/101/25)` `=(50+52)/50` =53/50

Math
You 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...

Math
Evaluate `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

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=sqrt(t^2+4)` and `g(x)=F_0(x).`...

Math
`F(x)=int_x^pisqrt(1+sec(t))dt` `F(x)=int_pi^xsqrt(1+sec(t))dt` `F'(x)=d/dxint_pi^xsqrt(1+sec(t))dt` `F'(x)=sqrt(1+sec(x))`

Math
You need to use the Part 1 of the FTC to evaluate the derivative of the function. You need to notice that the function G(x) is the composite of two functions `f(x) = int_1^x cos t dt ` and `g(x) =...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=ln(t)` and `h(x)=F_1(e^x).` Therefore...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=t^2/(t^4+1)` and `h(x)=F_1(sqrt(x))`...

Math
You need to use the Part 1 of the FTC to evaluate the derivative of the function. You need to notice that the function y is the composite of two functions `f(x) = int_1^x sqrt(1+sqrt t) dt` and...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=cos^2(t)` and `y(x)=F_0(x^4).` Therefore...

Math
`y=int_(13x)^1(u^3/(1+u^2))du` Let t=13x `dt/dx=3` `dy/dx=dy/dt*dt/dx` `y'=d/dxint_(13x)^1(u^3/(1+u^2))du` `=d/dtint_t^1(u^3/(1+u^2))du.dt/dx` `=d/dtint_1^t(u^3/(1+u^2))du.dt/dx`...

Math
You need to use the Part 1 of the FTC to evaluate the derivative of the function. You need to notice that the function h(x) = y is the composite of two functions `f(x) = int_1^x sqrt(1+t^2)dt` and...

Math
You 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)...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=1/(t^3+1)` and `g(x)=F_1(x).` Therefore...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=e^(t^2t)` and `g(x)=F_3(x).` Therefore...

Math
Hello! Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f` `F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.` Here `f(t)=(tt^2)^8` and `g(x)=F_5(x).` Therefore...

Math
To solve, assign variables that represent the number of peanuts in each bowl. Let x be the original number of peanuts in Bowl A. And let y be the original number of peanuts in Bowl B. If 17 peanuts...

Math
`24+68+1012+14` `...+210` To compute this, group the positive numbers and group the negative numbers together. In getting the sum of the negative numbers, consider the last negative term. `=...

Math
To take the sum of i and i, the operation that should be performed is addition. `i + i` To add like terms, add the coefficients and copy the variable. Since there are no written numbers at the...

Math
We are asked to find the circumcenter and orthocenter for triangle ABC with vertices at A(5,7),B(0,2) and C(1,1). (1) The circumcenter is the intersection of the perpendicular bisectors of the...

Math
You need to use the mean value theorem such that: `int_a^b f(x)dx = (ba)f(c), c in (a,b)` `int_(1)^1 sqrt(1+x^2)dx = (1+1)f(c) = 2f(c)` You need to verify the monotony of the function `f(x) =...

Math
You need to use the mean value thorem to verify the given inequality, such that: int_a^b f(x)dx = (ba)*f(c), c in (a,b) Replacing cos x for f(x) and pi/6 for a, pi/4 for b, yields:...

Math
`int_3^0(1+sqrt(9x^2))dx` Consider the graph of y=f(x)=`1+sqrt(9x^2)` `y=1+sqrt(9x^2)` `y1=sqrt(9x^2)` `(y1)^2=9x^2` `x^2+(y1)^2=3^2` This is the equation of circle of radius 3 centred at...

Math
`int_5^5(xsqrt(25x^2))dx` `=int_5^5xdxint_5^5sqrt(25x^2)dx` `=I_1I_2` I_1 can be be interpreted as area of two triangles;one above the xaxis and the other below axis.Since they are on the...

Math
`int_(1)^2 x dx` To interpret this in terms of area, graph the integrand. The integrand is the function f(x) =x. Then, shade the region bounded by the graph of f(x)=x and the xaxis in the...

Math
`int_0^10 x5dx` To interpret this in terms of area, graph the integrand. The integrand is the function f(x) = x  5. Then, shade the region bounded by f(x) = x5 and the xaxis in the...

Math
You need to use the mean value thorem to verify the given inequality, such that: `int_a^b f(x)dx = (ba)*f(c), c in (a,b)` Replacing `x^2  4x + 4` for `f(x)` and 0 for a, 4 for b, yields: `int_0^4...

Math
You need to check if` int_0^1 sqrt(1+x^2)dx <= int_0^1sqrt(1+x)dx` , using mean value theorem, such that: `int_a^b f(x)dx = (ba)f(c), ` where `c in (a,b)` `int_0^1 sqrt(1+x^2)dx <=...

Math
You need to use the midpoint rule to approximate the interval. First, you need to find `Delta x` , such that: `Delta x = (ba)/n` The problem provides b=8, a=0 and n = 4, such that: `Delta x =...