
Math
We have to be careful here because of absolute value sign. We need to know where the function under absolute value changes sign (in the interval of integration), which is at `x=pi` in this case....

Math
Hello! Find the indefinite integral: `int((sqrt(y)y)/y^2)dy=int(y^(3/2)1/y)dy=(2)*y^((1)/(2))ln(y).` Then substitute y from 1 to 4: `(2*4^(1/2)ln(4))(2)=1ln(4)+2=1ln(4) approx 0.386.`

Math
Hello! First find the indefinite integral, `int(x/(2)2/x)dx=int(x/2)dxint(2/x)dx=(x^2)/42ln(x).` Then substitute x from 1 to 2: `(2^2/42ln(2))(1^2/42ln(1))=12ln(2)1/4=3/42ln(2) approx...

Math
You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_0^1(5x  5^x)dx = int_0^1(5x)dx  int_0^1 5^x dx`...

Math
You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_0^1(x^10 + 10^x)dx = int_0^1(x^10)dx + int_0^1 10^x...

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_(pi/4)^(pi/3) csc^2 theta d theta = int_(pi/4)^(pi/3) 1/(sin^2...

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_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta =...

Math
We will make substitution `x=t^6.` Therefore, the differential is `dx=6t^5dt` and the new bounds of integration are `t_1=root(6)(1)=1` and `t_2=root(6)(64)=2.` `int_1^64(1+root(3)(x))/sqrt x...

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)` `You need to expand the cube such that: (x1)^3 = x^3  1  3x(x1)`...

Math
Hello! `(t^21)/(t^41)=1/(t^2+1).` Therefore indefinite integral is `arctan(t)(+C).` Substitute t from 0 to `1/sqrt(3)` and obtain `arctan(1/sqrt(3))arctan(0)=pi/60=pi/6 approx 0.524.`

Math
We have to be careful because of the absolute value. We need to know where the function under absolute values is positive and when it is negative in the given interval. It is also a good idea to...

Math
Hello! Consider two intervals: (1, 0) and (0, 2) to simplify x: `int_(1)^2(x2x)dx=int_(1)^0(x2x)dx+int_0^2(x2x)dx=` `=int_(1)^0(x+2x)dx+int_0^2(x2x)dx=`...

Math
You need to find the indefinite integral, hence, you need to remember that `sec t = 1/(cos t)` , such that: `int (sec t)(sec t + tan t) dt = int (1/(cos t))(1 + sin t)/(cos t) dt` `int (sec t)(sec...

Math
You need to find the indefinite integral, hence, you need to remember that `1 + tan^2 x = 1/(cos^2 x) = (tan x)'.` `int (1 + tan^2 x )dx = int (tan x)' dx = tan x + c` Hence, evaluating the...

Math
You need to find the indefinite integral, hence, you need to remember that sin `2x = 2sin x*cos x` , such that: `int (sin 2x)(sin x) dx = int (2sin x*cos x)/(sin x) dx ` `int (sin 2x)(sin x) dx =...

Math
You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_(2)^3(x^3  3)dx = int_(2)^3(x^3)dx  int_(2)^3 3...

Math
You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that: `int_a^b f(x)dx = F(b)  F(a)` `int_1^2(4x^3  3x^2 + 2x)dx = int_1^2(4x^3)dx  int_1^2...

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_(2)^0 ((1/2)t^4 + (1/4)t^3  t)dt = int_(2)^0...

Math
We will use linearity of integral: `int(a cdot f(x)+b cdot g(x))dx=a int f(x)dx+b int g(x)dx.` `int_0^3(1+6w^210w^4)dw=int_0^3dw+6int_0^3w^2dw10int_0^3w^4dw=`...

Math
You need to use the following substitution to evaluate the definite integral, such that: `1  t = u => dt = du` `int_(1)^1 t*(1t)^2dt = int_(u_1)^(u_2) (1  u)*u^2 (du)` `int_(u_1)^(u_2) (u...

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_0^pi (5e^x+ 3sin x)dx = int_0^pi 5e^x dx + int_0^pi 3sin x...

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_1^2 (1/x^2  4/x^3)dx = int_1^2 1/x^2 dx  int_1^2 4/x^3 dx`...

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_0^4(3sqrt t  2e^t)dt = int_0^4 3sqrt tdt  int_0^4 2e^t...

Math
You 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, `...

Math
You 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
You 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`...

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_0^3 (2sin x  e^x) dx =int_0^3 2sin x dx  int_0^3 e^x dx`...

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
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
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
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
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
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,...