Self Improvement Tips & Resources

1) ∫(3x^2+4x^2-7)dx.
2)∫ (2x-1)(x^2-x^6)dx.
3)∫ x-1/4x^2-8x+10dx
4)∫(x+1)^2/x dx.
5)∫ 3e^2n dx,
6)∫ from 1to2 x^2e^x^3dx.

please help me solve I am studying Mathematics "self study"

1: simple polynomial calculus.

first, simplify integrand:
∫(7x^2 - 7)dx

Next, use the addition rule, constant multiplication rule, and power rule to do the simplest integration you can do:

∫(7x^2 - 7)dx = 7/3 * x^3 - 7*x + C

remember the C when doing indefinite integrals, you will loose points should you not.
————————————-
2: ∫ (2*x-1)*(x^2-x^6)dx.

Expand integrand and sort by power:
∫(-2*x^7 + x^6 + 2*x^3 - x^2) dx

Carry out integration as you normally would:
∫(-2*x^7 + x^6 + 2*x^3 - x^2) dx =

-1/4*x^8 + 1/7*x^7 + 1/2*x^4 - 1/3*x^3 + C
————————————–…
3) ∫(x-1/4 * x^2 - 8*x+10)dx

Still simple polynomial calculus. First sort by power and combine terms which can be combined:
∫(-1/4 * x^2 - 7*x + 10)dx =

-1/12*x^3 - 7/2*x^2 + 10*x + C
————————————–…
4: ∫(x+1)^2/x dx

Expand integrand, and then do the simple polynomial calculus. The 1/x is a little tricky, but recall that 1/x is the same as x^(-1). When you apply the power rule, it gets interesting because your term will have a 1/0 coefficient, which is against the rules. By a coincidence in mathematics, ∫(1/x)dx = ln(x) + C

∫(x + 2 + 1/x) dx =

1/2*x^2 + 2*x + ln(x) + C
————————————–
5: ∫ 3*e^(2*n) dx, if I understand you correctly. You might want to re-learn to use parenthesis and order of operations.

This is a trick question. n is a variable, but isn’t the variable of integration. Unless you tell me how n is a function of x, but since you didn’t, I must assume n is a constant.

Therefore the answer is
3*x*e^(2*n) + C
————————————-
6. int(x^2 * e^(x^3), x= 1..2), if I understand you correctly

Use the trick of u-substitution to solve ∫(x^2 * e^(x^3)) dx.

Define u = x^3 and take its derivative du = 3*x^2 dx

Can we spot the derivative in the original integral? yes, if we multiply by 1 in a fancy way (ahem, 3/3).

∫(x^2 * e^(x^3)) dx = ∫(3/3 * x^2 * e^(x^3)) dx

since du = 3*x^2 dx, replace this term with du. Replace our definition of u in the exponent. Factor the 1/3 out of the integration.

1/3 * ∫e^u du

∫e^u du is easy, it’s just equal to e^u + C. The natural exponential has this nice property.

Therefore, including the definition of u and the 1/3 factor, the final result of the indefinite integral is
1/3 * e^(x^3) + C

Plug in the limits:
int(x^2 * e^(x^3), x= 1..2) = [1/3 * e^((2)^3)] - [1/3 * e^((1)^3)]

Result:
1/3 * e^8 - e/3