
Math
`f''(t)=2e^t+3sin(t)` `f'(t)=int(2e^t+3sin(t))dt` `f'(t)=2e^t3cos(t)+C_1` `f(t)=int(2e^t3cos(t)+C_1)dt` `f(t)=2e^t3sin(t)+C_1t+C_2` Now let's find constants C_1 and C_2 , given f(0)=0 and...

Math
`f''(x)=2/3x^(2/3)` `f'(x)=int2/3x^(2/3)dx` `f'(x)=2/3(x^(2/3+1)/(2/3+1))+c_1` `f'(x)=(2/3)(3/5)x^(5/3)+c_1` `f'(x)=2/5x^(5/3)+c_1` c_1 is constant `f(x)=int(2/5x^(5/3)+c_1)dx`...

Math
You need to evaluate f, knowing the second derivative, hence, you need to use the following principle, such that: `int f''(x)dx = f'(x) + c` `int f'(x) dx = f(x) + c` Hence, you need to evaluate...

Math
`f'''(t)=cos(t)` `f''(t)=intf'''(t)dt` `f''(t)=intcos(t)dt` `f''(t)=sin(t)+c_1` `f'(t)=int(sin(t)+c_1)dt` `f'(t)=cos(t)+c_1t+c_2` `f(t)=intf'(t)dt` `f(t)=int(cos(t)+c_1t+c_2)dt` `...

Math
`f'''(t)=e^t+t^(4).` Integrate this once: `f''(t)=e^t+(1/3)t^(3)+C_1,` twice: `f'(t)=e^t+(1/2)(1/3)t^(2)+C_1t+C_2,` thrice: `f(t)=e^t+(1/1)(1/2)(1/3)t^(1)+C_1t^2+C_2t+C_3=...

Math
`f'(x)=1+3sqrt(x)` `f(x)=int(1+3sqrt(x))dx` `f(x)=x+3(x^(1/2+1)/(1/2+1))+c` `f(x)=x+3(x^(3/2)/(3/2))+c` `f(x)=x+3(2/3)x^(3/2)+c` `f(x)=x+2x^(3/2)+c` Now let's find constant c , given f(4)=25...

Math
You need to evaluate f and the problem provides f'(x), hence, you need to use the following relation, such that: `int f'(x)dx = f(x)+ c` `int (5x^4  3x^2 + 4)dx = f(x) + c` You need to evaluate...

Math
The general antiderivative of f' is `f(t)=4arctan(t)+C,` where C is any constant. It must be f(1)=0, so C=4arctan(1). acrctan(1) is `pi/4,` so the answer is `f(t)=4arctan(t)pi.`

Math
Integrating gives `f(t)=t^2/2(1/2)t^(2)+C,` where C is any constant. `f(1)=1/21/2+C=C=6,` so C=6. The answer: `f(t)=t^2/21/(2t^2)+6.`

Math
You need to evaluate the antiderivative of the function f'(t), such that: `int f'(t)dt = f(t) + c` `int (2cos t + sec^2 t) dt = int 2cos t dt + int sec^2 t dt` `int (2cos t + sec^2 t) dt = 2sin t +...

Math
You need to evaluate the antiderivative of the function f'(x), such that: `int f'(x)dx = f(x) + c` `int (x^21)/x dx = int x^2/x dx  int 1/x dx` `int (x^21)/x dx = int x dx  int 1/x dx` `int...

Math
You need to evaluate f and the problem provides f'(x), hence, you need to use the following relation, such that: `int f'(x)dx = f(x)+ c` You need to evaluate the indefinite integral of the power...

Math
`f'(x)=4/sqrt(1x^2)` `f(x)=int(4/sqrt(1x^2))dx` `f(x)=4arcsin(x)+c_1` Let's find constant c_1 , given f(1/2)=1 `f(1/2)=1=4arcsin(1/2)+c_1` `1=4(pi/6)+c_1` `c_1=1(2pi)/3`...

Math
`f''(x)=2+12x12x^2` `f'(x)=int(2+12x12x^2)dx` `f'(x)=2x+12(x^2/2)12(x^3/3)+c_1` `f'(x)=2x+6x^24x^3+c_1` Now let's find constant c_1 , given f'(0)=12 `f'(0)=12=2(0)+6(0^2)4(0^3)+c_1`...

Math
`f''(x)=8x^3+5` `f'(x)=int(8x^3+5)dx` `f'(x)=8(x^4/4)+5x+c_1` `f'(x)=2x^4+5x+c_1` Now let's find constant c_1 , given f'(1)=8 `f'(1)=8=2(1)^4+5(1)+c_1` `8=2+5+c_1` `c_1=1` `:.f'(x)=2x^4+5x+1`...

Math
`f(x) =1/52/x` To determine the most general antiderivative of this function, take the integral of it. `F(x)=int f(x)dx = int(1/52/x)dx` Then, apply the integral formula `int cdu=cu +C` and `int...

Math
The most general antiderivative F(t) of the function f(t) can be found using the following relation: `int f(t)dt = F(t) + c` `int (3t^4  t^3 + 6t^2)/(t^4)dt = int (3t^4)/(t^4)dt  int...

Math
The most general antiderivative G(t) of the function g(t) can be found using the following relation: `int g(t)dt = G(t) + c` `int (1 + t + t^2)/(sqrt t)dt = int (1)/(sqrt t)dt + int (t)/(sqrt t)dt...

Math
The most general antiderivative `R(theta)` of the function `r(theta)` can be found using the following relation: `int r(theta)d theta = R(theta) + c` `int (sec theta*tan theta  2 e^theta)d theta =...

Math
The most general antiderivative `H(theta)` of the function `h(theta)` can be found using the following relation: `int h(theta)d theta = H(theta) + c` `int (2sin theta  sec^2 theta)d theta = int...

Math
`int f(t) dt=int sin(t)+2 sinh(t) dt = int sin(t) dt + 2 int sinh(t) dt ` We note that the derivative of `cosh(t)=sinh(t) =>d/dx cosh(t) = sinh(t)=>d/dx sinh(t)=cosh(t)` This means that `int...

Math
Integrating f gives (both integrals are wellknown) `5e^x3sinh(x)+C,` where `C` is any constant. This is the answer. Checking by differentiation:...

Math
You need to evaluate the most general antiderivative, using the following rule, such that: `int f(x) dx = F(x) + c` `int (2sqrtx + 6cos x) dx = int 2sqrt x dx + int 6cos x dx` `int (2sqrtx + 6cos...

Math
You need to evaluate the most general antiderivative, using the following rule, such that: `int f(x) dx = F(x) + c` `int (x^5  x^3 + 2x)/(x^4) dx = int(x^5)/(x^4) dx  int (x^3)/(x^4)dx + int...

Math
You need to evaluate the most general antiderivative, using the following rule, such that: `int f(x) dx = F(x) + c` `int (2 +x^2)/(1+x^2) dx = int(1+x^2)/(1+x^2) dx + int 1/(1+x^2)dx ` `int (2...

Math
You need to evaluate the antiderivative F,under the given condition, such that: `int f(x)dx = F(x) + c` Hence, you need to evaluate the indefinite integral of function f(x), such that: `int (5x^4 ...

Math
`f(x)=43(1+x^2)^1` `F=int(43(1+x^2)^1)dx` `F=int4dxint3/(1+x^2)dx` `F=4x3arctan(x)+c_1` Let's find constant c_1 , given F(1)=0 `F(1)=0=4(1)3arctan(1)+c_1` `0=43(pi/4)+c_1` `c_1=(3pi)/44`...

Math
`f''(x)=20x^312x^2+6x` `f'(x)=int(20x^312x^2+6x)dx` Applying the sum rule and power yields, `f'(x)=20(x^4/4)12x^3/3+6x^2/2+c_1` `f'(x)=5x^44x^3+3x^2+c_1` `f(x)=intf'(x)dx`...

Math
`f''(x)=x^64x^4+x+1` `f'(x)=intf''(x)dx` `f'(x)=int(x^64x^4+x+1)dx` `f'(x)=x^7/74(x^5/5)+x^2/2+x+c_1` `f'(x)=x^7/7(4x^5)/5+x^2/2+x+c_1` `f(x)=intf'(x)dx`...

Math
`f(x) = x  3` To find the general antiderivative of this function, we take its integral. `F(x) = int f(x) dx = int(x3)dx` Take the integral of each term. `=intxdx  int3dx` For the first...

Math
You need to evaluate the most antiderivative of the function, such that: `int f(x) dx = F(x) + c` `int ((1/2)x^2  2x + 6) dx = int(1/2)x^2 dx  int 2xdx+ int 6 dx` `int ((1/2)x^2  2x + 6) dx =...

Math
`f(x)=1/2+3/4x^24/5x^3 ` To determine the most general antiderivative of this function, take the integral of f(x). `F(x)=intf(x) dx= int(1/2+3/4x^24/5x^3)dx` Then, apply the integral formula `int...

Math
You need to evaluate the most antiderivative of the function, such that: `int f(x) dx = F(x) + c` `int (8x^9  3x^6 + 12x^3) dx = int8x^9 dx  int 3x^6dx+ int12x^3 dx` `int (8x^9  3x^6 + 12x^3) dx...

Math
First, you need to open the brackets, such that: `f(x) = (x+1)(2x1) = 2x^2 + x  1` You need to evaluate the most antiderivative of the function, such that: `int f(x) dx = F(x) + c` `int (2x^2 + x...

Math
You need to evaluate the most general antiderivative, using the following rule, such that: `int f(x) dx = F(x) + c` First, you need to open the brackets, such that: `f(x) = x(2x)^2 = x*(4  4x +...

Math
The most general antiderivative F(x) of the function f(x) can be found using the following relation: `int f(x)dx = F(x) + c` `int (7x^(2/5) + 8x^(4/5))dx = int (7x^(2/5))dx + int (8x^(4/5))dx`...

Math
The most general antiderivative F(x) of the function f(x) can be found using the following relation: `int f(x)dx = F(x) + c` `int (x^(3.4)  2x^(sqrt2  1))dx = int (x^(3.4))dx  int (2x^(sqrt2 ...

Math
The answer is `F(x) = x*sqrt(2) + C,` where C is any constant. Let's check: `F'(x) = (x*sqrt(2) + C)' = x'*sqrt(2) + 0 = 1*sqrt(2) = sqrt(2),` QED.

Math
`e^2` isn't depend on x, so its antiderivative is `x*e^2.` The general antiderivative is `x*e^2+C,` where C is any constant. Let's check: `(x*e^2+C)'=x'*e^2+C'=1*e^2+0=e^2.`

Math
The most general antiderivative F(x) of the function f(x) can be found using the following relation: `int f(x)dx = F(x) + c` `int (3sqrt x  2root(3) x)dx = int (3sqrt x)dx  int (2root(3) x)dx`...

Math
`f(x)= root(3)(x^2) + xsqrtx` To determine the most general antiderivative of this function, take the integral of f(x). `F(x) = intf(x) dx = int (root(3)(x^2)+xsqrtx)dx` Express the radicals in...

Math
Hello! At first, consider line graph. It gives the same information as the bar graph, but line graph combines data from both cities which makes comparing easier. From the line graph we see those...

Math
Votes received by Troy Smith=9/10 of votes Votes received by Darren =1/20 of votes Votes received by Brady = 1/50 of votes Now let's add the combined votes of Troy Smith , Darren and Brady in...

Math
There are an infinite number of values on the number line between `1/3` and ```1/4` .`` One way to find a value between is to write equivalent fractions using a common denominator. Because the...

Math
Hello! When two points of the straight line are given, the slope of this line is `(y_2y_1)/(x_2x_1).` Here this is equal to `(28)/(r6) = 10/(r6)` and must be 1. So `10/(r6)=1,` r6=10,...

Math
Hello! Exponential growth function has a form `y=A*(1+r)^x,` where A is the initial value and r is the growth rate. If we substitute x=0 we obtain `y(0)=A*(1+r)^0=A.` Recall that yintercept is the...

Math
The question is based on the similarity of triangles. In similar triangles ratio of the corresponding sides is same. So let's find the scale factor Scale factor=2/1.5 = 1.33 So,...

Math
The yintercept of a graph of the function is the ycoordinate of the point where the graph intersects yaxis. The xcoordinate of this point is 0. Therefore, to find the yintercept, plug in x = 0...

Math
Hello! Let's denote the number of minivans washed as m and the number of cars as c. Then she earned 7m dollars for minivans and 4c for cars. It is given that `7m+4c=41.` If m and c could be...

Math
t(1)=4 t(2)=4+4/5=24/5 t(3)=4+4/5+4/25=124/25 t(4)=4+4/5+4/25+4/125=624/125 For each nth term we take t(1) times r^(n1): e.g. when n=4 we have 4(1/5)^3=4/125...