Integrate the following: i) Find ∫ (4x2 + 4x + 4) dx ii) Using Special integrals Find: ∫ dx / (3x2 + 4x + 2)

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Here's the completion of the previous problem and the solution for the new one.

1) Integrate the following

i) Find $\int (4x^2 + 4x + 4) dx$

Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , to each term.
$\int (4x^2 + 4x + 4) dx = 4\frac{x^{2+1}}{2+1} + 4\frac{x^{1+1}}{1+1} + 4x + C$
Simplify the expression.
$= 4\frac{x^3}{3} + 4\frac{x^2}{2} + 4x + C$ $= \frac{4}{3}x^3 + 2x^2 + 4x + C$
The final answer is $\boxed{\frac{4}{3}x^3 + 2x^2 + 4x + C}$ .

2) Using Special integrals Find:

i) $\int \frac{dx}{3x^2 + 4x + 2}$
Complete the square in the denominator.
$3x^2 + 4x + 2 = 3\left(x^2 + \frac{4}{3}x + \frac{2}{3}\right)$ $= 3\left(x^2 + \frac{4}{3}x + \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2 + \frac{2}{3}\right)$ $= 3\left(\left(x + \frac{2}{3}\right)^2 - \frac{4}{9} + \frac{6}{9}\right)$ $= 3\left(\left(x + \frac{2}{3}\right)^2 + \frac{2}{9}\right)$
Rewrite the integral using the completed square form.
$\int \frac{dx}{3\left(\left(x + \frac{2}{3}\right)^2 + \frac{2}{9}\right)} = \frac{1}{3} \int \frac{dx}{\left(x + \frac{2}{3}\right)^2 + \left(\frac{\sqrt{2}}{3}\right)^2}$
Apply the special integral formula $\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$ .
Here, $u = x + \frac{2}{3}$ (so $du = dx$ ) and $a = \frac{\sqrt{2}}{3}$ .
Substitute $u$ and $a$ into the formula and simplify.
$\frac{1}{3} \cdot \frac{1}{\frac{\sqrt{2}}{3}} \arctan\left(\frac{x + \frac{2}{3}}{\frac{\sqrt{2}}{3}}\right) + C$ $= \frac{1}{3} \cdot \frac{3}{\sqrt{2}} \arctan\left(\frac{\frac{3x+2}{3}}{\frac{\sqrt{2}}{3}}\right) + C$ $= \frac{1}{\sqrt{2}} \arctan\left(\frac{3x+2}{\sqrt{2}}\right) + C$
Rationalize the denominator.
$= \frac{\sqrt{2}}{2} \arctan\left(\frac{3x+2}{\sqrt{2}}\right) + C$
The final answer is $\boxed{\frac{\sqrt{2}}{2} \arctan\left(\frac{3x+2}{\sqrt{2}}\right) + C}$ .

Integrate the following: i) Find ∫ (4x2 + 4x + 4) dx ii) Using Special integrals Find: ∫ dx / (3x2 + 4x + 2)

Step-by-step solution

Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , to each term.

Simplify the expression.

Complete the square in the denominator.

Rewrite the integral using the completed square form.

Apply the special integral formula $\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$ .

Substitute $u$ and $a$ into the formula and simplify.

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More Mathematics Questions

Integrate the following: i) Find ∫ (4x2 + 4x + 4) dx ii) Using Special integrals Find: ∫ dx / (3x2 + 4x + 2)

Step-by-step solution

Apply the power rule for integration, ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C, to each term.

Simplify the expression.

Complete the square in the denominator.

Rewrite the integral using the completed square form.

Apply the special integral formula ∫1u2+a2du=1aarctan⁡(ua)+C\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C∫u2+a21​du=a1​arctan(au​)+C.

Substitute uuu and aaa into the formula and simplify.

Still stuck on this one?

More Mathematics Questions

Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , to each term.

Apply the special integral formula $\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$ .

Substitute $u$ and $a$ into the formula and simplify.