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MathYou 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...

MathWe 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=`...

MathYou 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...

MathYou 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...

MathYou 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`int_1^4(4+6u)/sqrt(u)du` `=int_1^4(4/sqrt(u)+(6u)/sqrt(u))du` `=int_1^4(4u^(1/2)+6u^(1/2))du` `=[4(u^(1/2+1)/(1/2+1))+6(u^(1/2+1)/(1/2+1))]_1^4` `=[4(u^(1/2)/(1/2))+6(u^(3/2)/(3/2))]_1^4`...

MathYou 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...

MathYou need to evaluate the definite integral, such that: `int_0^1 x(root(3) x + root(4) x)dx = int_0^1 (x^(1+1/3) + x^(1+1/4))dx` `int_0^1 x(root(3) x + root(4) x)dx = ((x^(2+1/3))/(2+1/3) +...

MathYou 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, `...

MathYou 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`intv(v^2+2)^2dv` `=intv((v^2)^2+2v^2*2+2^2)dv` `=intv(v^4+4v^2+4)dv` `=int(v^5+4v^3+4v)dv` apply the power rule, `=v^6/6+4v^4/4+4v^2/2+C` , C is constant `=v^6/6+v^4+2v^2+C`

Math`int(x^32sqrt(x))/xdx` ``Simplify by dividing each term in the numerator by x. ` <br> ` `=(x^3/3)(2x^(1/2))/(1/2)+C` `=(x^3/3)4x^(1/2)+C` `` ` `

Math`int(x^2+1+1/(x^2+1))dx` apply the sum rule, `=intx^2dx+int1dx+int1/(x^2+1)dx` To evaluate the above integrals, we know that, `intx^ndx=x^(n+1)/(n+1)` and `int1/(x^2+1)dx=arctan(x)` using above,...

Math`int(csc^2(t)2e^t)dt` Apply the sum rule, `=intcsc^2(t)dtint(2e^tdt` We now the following common integrals, `intcsc^2(x)=cot(x)` and `inte^xdx=e^x` evaluate using the above, `=cot(t)2e^t+C`...

MathYou 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`g(x)=int_(2x)^(3x)(u^21)/(u^2+1)du` `g(x)=int_(2x)^0(u^21)/(u^2+1)du+int_0^(3x)(u^21)/(u^2+1)du` `g(x)=int_0^(2x)(u^21)/(u^2+1)du+int_0^(3x)(u^21)/(u^2+1)du`...

Math`F(x)=int_x^(x^2)(e^(t^2))dt` From the fundamental theorem of calculus, `int_x^(x^2)(e^(t^2))dt=F(x^2)F(x)` `d/dxint_x^(x^2)(e^(t^2))dt=F'(x^2).d/dx(x^2)F'(x)` `=2x(e^((x^2)^2))e^(x^2)`...

MathYou 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`int_1^2((v^3+3v^6)/v^4)dv` simplify the integrand and apply the sum rule, `=int_1^2(v^3/v^4+(3v^6)/v^4)dv` `=int_1^2(1/v+3v^2)dv` using the following common integrals `int1/xdx=ln(x)` and...

MathEvaluate `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...

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...

MathYou 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)...

MathYou 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)` ....

MathYou 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`...

MathEvaluate `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

MathEvaluate `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`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

MathYou 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...

MathEvaluate `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

MathYou 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)...

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