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MathFor the given problem: `int_0^1 xe^(x^2)` , we may first solve for its indefinite integral. Indefinite integral are written in the form of `int f(x) dx = F(x) +C` where:` f(x)` as the integrand...

MathIndefinite integral follows the formula: `int f(x) dx = F(x)+C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x) ` `C` as constant of integration. The given integral...

Math`inte^(3x)/(1+e^x)^3dx` `=int((e^x)^2e^x)/(1+e^x)^3dx` Now apply integral substitution:`u=e^x` `=>du=e^xdx` `=intu^2/(1+u)^3du` Now use the following from the integration tables:...

MathRecall that indefinite integral follows: `int f(x) dx = F(x)+C` where: `f(x)` as the integrand function `F(x) ` as the antiderivative of `f(x)` `C` as constant of integration. To evaluate given...

MathRecall that indefinite integral follows `int f(x) dx = F(x) +C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C ` as the constant of integration The given integral...

MathIndefinite integral follows the formula: `int f(x) dx = F(x)+C` where: `f(x)` as the integrand function `F(x) ` as the antiderivative of `f(x)` `C` as constant of integration. The given integral...

Math`intx/(x^26x+10)^2dx` Let's rewrite the integrand as, `=1/2int(2x)/(x^26x+10)^2dx` `=1/2int(2x6+6)/(x^26x+10)^2dx` `=1/2[int(2x6)/(x^26x+10)^2dx+int6/(x^26x+10)^2dx]`...

MathIndefinite integral follows the formula: `int f(x) dx = F(x)+C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C ` as constant of integration. To evaluate the given...

Math`intln(x)/(x(3+2ln(x)))dx` First let's apply integral substitution: `u=lnx` `=>du=1/xdx` `=intu/(3+2u)du` From the integration tables, `intu/(a+bu)du=1/b^2(bualna+bu)+C` Using the above,...

MathFor the given integral problem: `int sqrt(x)arctan(x^(3/2))dx` , we can evaluate this applying indefinite integral formula: `int f(x) dx = F(x) +C` . where: `f(x)` as the integrand function `F(x)`...

MathRecall that indefinite integral follows `int f(x) dx = F(x) +C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C` as the constant of integration. The given integral...

MathRecall that indefinite integral follows `int f(x) dx = F(x) +C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C` as the constant of integration. The given integral...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x) ` as the integrand `F(x) ` as the antiderivative function `C` as the arbitrary constant...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x)` as the antiderivative function `C` as the arbitrary constant...

Math`intx/(1sec(x^2))dx` First , let's apply integral substitution:`u=x^2` `=>du=2xdx` `=int1/(1sec(u))(du)/2` Take the constant out, `=1/2int1/(1sec(u))du` From the integration tables,...

Math`inte^x/(1tan(e^x))dx` First, let's apply integral substitution:`u=e^x` `=>du=e^xdx` `=int1/(1tan(u))du` Now from the integration tables: `int1/(1+tan(u))du=1/2(u+lncos(u)+sin(u))+C`...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x)` as the antiderivative function `C` as the arbitrary constant...

Math`inttheta^3/(1+sin(theta^4))d theta` First let's apply integral substitution:`u=theta^4` `=>du=4theta^3d theta` `=int1/(1+sin(u))(du)/4` Take the constant out, `=1/4int1/(1+sin(u))du` From the...

Math`int(4x)/(25x)^2dx` Take the constant out, `=4intx/(25x)^2dx` From the integration tables, use the following, `intu/(a+bu)^2du=1/b^2(a/(a+bu)+lna+bu)+C` Here we have `a=2,b=5`...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x) ` as the antiderivative function `C ` as the arbitrary constant...

MathRecall that indefinite integral follows the formula: `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x)` as the antiderivative function `C` as the arbitrary...

MathFrom the table of integrals, we have a integration formula for inverse sine function as: `int arcsin(u/a)du = u*arcsin(u/a) +sqrt(a^2u^2) +C` It resembles the given integral problem: `int...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x)` as the antiderivative function `C` as the arbitrary constant...

Math`int1/(sqrt(x)root(3)(x))dx` Apply integral substitution:`u=x^(1/6)` `=>du=1/6x^(1/61)dx` `du=1/6x^(5/6)dx` `du=1/(6x^(5/6))dx` `6x^(5/6)du=dx` `6(x^(1/6))^5du=dx` `6u^5du=dx`...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where: `f(x)` as the integrand `F(x)` as the antiderivative function of `f(x)` `C` as the arbitrary...

Math`inte^x/((e^(2x)+1)(e^x1))dx` Apply integral substitution:`u=e^x` `=>du=e^xdx` `=int1/((u^2+1)(u1))du` Now let's create partial fraction template for the integrand,...

Math`inte^x/((e^x1)(e^x+4))dx` Let's apply integral substitution:`u=e^x` `=>du=e^xdx` `=int1/((u1)(u+4))du` Now create partial fraction template of the integrand,...

Math`int(sec^2(x))/(tan(x)(tan(x)+1))dx` Let's apply integral substitution: `u=tan(x)` `du=sec^2(x)dx` `=int1/(u(u+1))du` Now let's create the partial fraction template for the integrand,...

Math`int(sec^2(x))/(tan^2(x)+5tan(x)+6)dx` Let's apply integral substitution:`u=tan(x)` `=>du=sec^2(x)dx` `=int1/(u^2+5u+6)du` Now we have to write down integrand as sum of partial fraction...

Math`int5cos(x)/(sin^2(x)+3sin(x)4)dx` Take the constant out, `=5intcos(x)/(sin^2(x)+3sin(x)4)dx` Now let's apply integral substitution:`u=sin(x)` `=>du=cos(x)dx` `=5int1/(u^2+3u4)du` Now to use...

Math`intsin(x)/(cos(x)+cos^2(x))dx` Apply integral substitution: `u=cos(x)` `=>du=sin(x)dx` `=int1/(u+u^2)(1)du` Take the constant out, `=1int1/(u+u^2)du` Now to compute the partial fraction...

Math`int (x^2+6x+4)/(x^4+8x^2+16)dx` To solve using partial fraction method, the denominator of the integrand should be factored. `(x^2+6x+4)/(x^4+8x^2+16) = (x^2+6x+4) / (x^2+4)^2` If the factor in...

MathIndefinite integral are written in the form of `int f(x) dx = F(x) +C` where:` f(x) ` as the integrand `F(x)` as the antiderivative function `C` as the arbitrary constant...

Math`int x/(16x^41)dx` To solve using partial fraction method, the denominator of the integrand should be factored. `x/(16x^41)=x/((2x1)(2x+1)(4x^2+1))` Take note that if the factors in the...

Math`int x^2/(x^42x^28)dx` To solve using partial fraction method, the denominator of the integrand should be factored. `x^2/(x^42x^28)=x^2/((x2)(x+2)(x^2+2))` If the factor in the denominator is...

MathFor the given integral problem: `int (6x)/(x^38)dx` , we may partial fraction decomposition to expand the integrand: `f(x)=(6x)/(x^38)` . The pattern on setting up partial fractions will depend...

Math`int(x^21)/(x^3+x)dx` `(x^21)/(x^3+x)=(x^21)/(x(x^2+1))` Now let's create partial fraction template, `(x^21)/(x(x^2+1))=A/x+(Bx+C)/(x^2+1)` Multiply equation by the denominator,...

Math`int(8x)/(x^3+x^2x1)dx` `(8x)/(x^3+x^2x1)=(8x)/((x^3+x^2)1(x+1))` `=(8x)/((x^2(x+1)1(x+1)))` `=(8x)/((x+1)(x^21))` `=(8x)/((x+1)(x+1)(x1))` `=(8x)/((x1)(x+1)^2)` Now let's form the partial...

MathFor the given integral problem: `int (x^3+3x4)/(x^34x^2+4x)dx` , we may simplify by applying long division since the highest degree of x is the same from numerator and denominator side....

Math`int(5x2)/(x2)^2dx` Let's use partial fraction decomposition on the integrand, `(5x2)/(x2)^2=A/(x2)+B/(x2)^2` `5x2=A(x2)+B` `5x2=Ax2A+B` comparing the coefficients of the like terms,...

Math`int (4x^2+2x1)/(x^3+x^2)dx` To solve using partial fraction method, the denominator of the integrand should be factored. `(4x^2+2x1)/(x^3+x^2)=(4x^2+2x1)/(x^2(x+1))` Take note that if the...

Math`int (x+2)/(x^2+5x) dx` To solve using partial fraction method, the denominator of the integrand should be factored. `(x+2)/(x^2+5x) = (x + 2)/(x(x+5))` Then, express it as sum of fractions....

Math`int(2x^34x^215x+5)/(x^22x8)dx` The integrand is a improper rational function,as the degree of the numerator is greater than the degree of the denominator.So we have to carry out division....

Math`int(x^3x+3)/(x^2+x2)dx` The given integrand is a improper rational function, as the degree of the numerator is more than the degree of the denominator. To apply the method of partial...

Math`int (x^2+12x+12)/(x^34x)dx ` To solve using partial fraction method, the denominator of the integrand should be factored. `(x^2+12x+12)/(x^34x) =(x^2+12x+12)/(x(x2)(x+2))` Then, express it as...

Math`int(3x)/(3x^22x1)dx` Let's use partial fraction decomposition on the integrand, `(3x)/(3x^22x1)=(3x)/(3x^2+x3x1)` `=(3x)/(x(3x+1)1(3x+1))` `=(3x)/((3x+1)(x1))` Now form the partial...

Math`int 5/(x^2+3x4)dx` To solve using partial fraction, the denominator of the integrand should be factored. `5/(x^2+3x4)=5/((x+4)(x1))` Then, express it as sum of fractions....

Math`int 2/(9x^21)` To solve using partial fraction method, the denominator of the integrand should be factored. `2/(9x^21) = 2/((3x1)(3x+1))` Then, express it as sum of fractions....

Math`int 1/(x^29)dx` To solve using the partial fraction method, the first step is to factor the denominator of the integrand. `1/(x^29) =1/((x  3)(x +3))` Then, express it as a sum of two...

MathTo find the arc length of a curve, we follow the formula: `S = int_a^b sqrt(1+((dy)/(dx))^2)` if `y=f(x)` , `alt=xlt=b` or ` [a,b]` . For the given problem: `y =x^2/2` on interval `[0,4]` , we...