Question

Evaluate the following integrals: (i) \(\int \frac{2 x^{3}+x^{2}+4}{\left(x^{2}+4\right)^{2}} d x\) (ii) \(\int \frac{x^{3}+x^{2}-5 x+15}{\left(x^{2}+5\right)\left(x^{2}+2 x+3\right)} d x\)(iii) \(\int \frac{d x}{\left(x^{4}+2 x+10\right)^{3}}\) (iv) \(\int \frac{x^{5}-x^{4}+4 x^{3}-4 x^{2}+8 x-4}{\left(x^{2}+2\right)^{3}} d x\)

Short answer

(i) Evaluate \(\int \frac{2 x^{3}+x^{2}+4}{\left(x^{2}+4\right)^{2}} d x\) Step 1: Integration by substitution Let \(u = x^2 + 4\). Then, \(\frac{du}{dx}=2x\). Now we can rewrite the integral in terms of \(u\): $$\frac{1}{2}\int\frac{2x^3+x^2+4}{u^2}\, du$$ Step 2: Simplify and integrate with respect to u Simplify the numerator and integrate with respect to \(u\): $$\frac{1}{2}\int\frac{u - 4}{u^2}\, du=-\frac{1}{2}\int\left(\frac{4}{u^2} - \frac{1}{u}\right)\,du=\frac{1}{2}\left(\int\frac{du}{u}-4\int\frac{du}{u^2}\right)$$ Now, integrate the resulting expression: $$\frac{1}{2} \left(\ln|u| + \frac{4}{u} \right) + C$$ Step 3: Substitute back and simplify Replace \(u\) with \(x^2 + 4\) and simplify the result: $$\frac{1}{2}\left(\ln|x^2 + 4| + \frac{4}{x^2 + 4}\right) + C$$ Please complete Steps 1-3 for each of the remaining problems (ii, iii, and iv) as outlined in the initial prompt.

Step-by-step solution

Integration by substitution

Let \(u = x^2 + 4\). Then, \(\frac{du}{dx} = 2x\). Now we can rewrite the integral in terms of \(u\): $$\frac{1}{2}\int\frac{2x^3+x^2+4}{u^2}\, du$$

Simplify and integrate with respect to u

Simplify the numerator and integrate with respect to \(u\): $$\frac{1}{2}\int\frac{u - 4}{u^2}\, du=-\frac{1}{2}\int\left(\frac{4}{u^2} - \frac{1}{u}\right)\,du=\frac{1}{2}\left(\int\frac{du}{u}-4\int\frac{du}{u^2}\right)$$ Now, integrate the resulting expression: $$\frac{1}{2} \left(\ln|u| + \frac{4}{u} \right) + C$$

Substitute back and simplify

Replace \(u\) with \(x^2 + 4\) and simplify the result: $$\frac{1}{2}\left(\ln|x^2 + 4| + \frac{4}{x^2 + 4}\right) + C$$ (ii) Evaluate \(\int \frac{x^{3}+x^{2}-5 x+15}{\left(x^{2}+5\right)\left(x^{2}+2 x+3\right)} d x\)

Partial fraction decomposition

Use partial fraction decomposition to rewrite this integral. We need to find constants A,B,C,D such that: $$\frac{x^3+x^2-5x+15}{(x^2+5)(x^2+2x+3)} = \frac{A}{x^2+5} + \frac{Bx+C}{x^2+2x+3}$$ Perform partial fraction decomposition and find A,B,C and D.

Evaluate the integral

Now we can rewrite the integral as follows: $$\int\frac{A}{x^2+5}\,dx + \int\frac{Bx+C}{x^2+2x+3}\,dx$$ Evaluate each integral and find an expression for the antiderivative.

Combine and simplify

Combine the results and simplify the expression. (iii) Evaluate \(\int \frac{d x}{\left(x^{4}+2 x+10\right)^{3}}\)

Integration by parts

To solve this integral, we can use integration by parts. Choose u(x) and dv(x) such that: $$u(x)=x^4+2x$$ and $$dv(x)=\frac{dx}{(x^4+2x+10)^2}$$ Find du/dx and integrate dv(x) to find v(x). Apply the integration by parts formula to get the desired integral.

Simplify the integral

Combine and simplify the expression for the integral. (iv) Evaluate \(\int \frac{x^{5}-x^{4}+4 x^{3}-4 x^{2}+8 x-4}{\left(x^{2}+2\right)^{3}} d x\)

Polynomial division

Perform polynomial division on the integrand to express the given integral as a polynomial plus a lower degree polynomial divided by \((x^2+2)^3\).

Evaluate the integral

Now we can rewrite the integral as the sum of two integrals: $$\int\text{polynomial}\,dx+\int\frac{\text{lower degree polynomial}}{(x^2+2)^{3}}\, dx$$ Evaluate each integral and find an expression for the antiderivative.

Combine and simplify

Combine the results and simplify the expression.