
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
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...

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_0^4(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

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
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
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
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
`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 =...

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=pi/2` , a=0 and n = 4, such that: `Delta x =...

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=2, a=0 and n = 5, such that: `Delta x =...

Math
You need to evaluate the definite integral using the mid point rule, hence, first you need to evaluate `Delta x:` `Delta x = (ba)/n => Delta x = (51)/4 = 1` You need to denote each of the 4...

Math
You need to use the fundamental theorem of calculus, to prove the equality, such that: `int_a^b f(x)dx = F(b)  F(a)` You need to replace x for f(x), such that: `int_a^b xdx = x^2/2_a^b` `int_a^b...

Math
You need to evaluate the definite integral, such that: `int_a^b f(x) dx = F(b)  F(a)` `int_a^b x^2 dx = (x^3)/3_a^b` `int_a^b x^2 dx = (b^3)/3  (a^3)/3` `int_a^b x^2 dx = (b^3  a^3)/3` Hence,...

Math
You have to recall the definition of the Reiman Integral `int_a^bf(x)dx=lim_(n>oo)sum_(i=1)^nf(x(i))Deltax` `where Deltax =(ba)/n and x(i)= a +iDeltax` `x ` `a=2 and b = 6 ` `Deltax = (62)/n=...

Math
`int_1^10 (x4ln(x))dx` To express this definite integral as limit of Riemann's Sum, apply the formula: `int_a^b f(x) dx = lim_(n> oo)sum_(i=1)^oo f(x_i)Delta x` where `Delta x = (ba)/n` `x_i...

Math
`int_1^2(1x)dx` To interpret the integral in terms of area , graph the integrand. The integrand is the function `f(x)=1x` Graph the function in the interval (1,2). Refer the attached graph. The...

Math
`int_0^9 (1/3x2)dx` To interpret this integral in terms of area, graph the integrand. The integrand is the function `f(x)=1/3x2` . Then, shade the region bounded by `f(x)=1/3x2` and the...

Math
`v(t)=2t1/(1+t^2)` position of the particle s(t) is given by, `s(t)=intv(t)dt` `s(t)=int(2t1/(1+t^2))dt` `s(t)=2(t^2/2)arctan(t)+C` , C is constant `s(t)=t^2arctan(t)+C` Now let's find C ,...

Math
`a(t) = sin(t) + 3cos(t)` `v(t) = cos(t) + 3sin(t) + a` `Now, v(0) = 2` `Thus, 2 = cos(0)+3sin(0)+a` `or, 2 = 1 + 0 + a` `or., a = 3` `Now, v(t) = cos(t) + 3sin(t) + 3` `Thus, s(t) = sin(t) ...