Problem 27

Question

Find the integral. $$ \int \frac{1}{4+4 x^{2}+x^{4}} d x $$

Step-by-Step Solution

Verified

Answer

$-\frac{1}{x^{2}+2} + C$

1Step 1: Substitution

Let $ u = x^{2}+2 $, then $ du = 2xdx $ and $ xdx = \frac{1}{2} du $. The equation $\int \frac{1}{4+4 x^{2}+x^{4}} d x$ will then become $\int \frac{1}{2u^{2}} du$

2Step 2: Apply the power rule for integrals

The power rule states that $\int u^{-n} du = \frac{1}{1-n}u^{1-n} + C$, where $C$ is the constant of integration. Now apply this rule to our integral, with $n=2$ we get $ \int u^{-2} du = \frac{1}{1-2}u^{1-2} + C = -u^{-1} + C$

3Step 3: Substitute back

Now substitute the substitution from step 1 back in: $-u^{-1} + C = -\frac{1}{u} + C = -\frac{1}{x^{2}+2} + C$

4Step 4: Final answer

To conclude, the integral $\int \frac{1}{4+4 x^{2}+x^{4}} d x$ equals to $-\frac{1}{x^{2}+2} + C$ where $C$ is the constant of integration.

Problem 27

Other exercises in this chapter

Problem 27

Find the integral involving secant and tangent. $$ \int \tan ^{2} x \sec ^{2} x d x $$

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Problem 27

In Exercises 27-30, verify the integration formula. $$ \int \frac{u^{2}}{(a+b u)^{2}} d u=\frac{1}{b^{3}}\left(b u-\frac{a^{2}}{a+b u}-2 a \ln |a+b u|\right)+C

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Problem 27

Use substitution to find the integral. $$ \int \frac{e^{x}}{\left(e^{x}-1\right)\left(e^{x}+4\right)} d x $$

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Problem 27

Solve the differential equation. $$ \frac{d y}{d t}=\frac{t^{2}}{\sqrt{2+3 t}} $$

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