如何运用多种积分技巧求解复杂积分？典型示例解析

Let’s walk through two classic integral problems that leverage trigonometric substitution, variable substitution, and integration by parts—essential tools for tackling tricky calculus integrals.

Example 1: $\int \sqrt{1-x^{2}} : dx$ #

This integral is ideal for trigonometric substitution thanks to the $\sqrt{1-x^2}$ term, which aligns with the Pythagorean identity $\sin^2(t) + \cos^2(t) = 1$.

• Trigonometric Substitution:
  Let $x = \sin(t)$. Then $dx = \cos(t)dt$, and substituting into the integral simplifies it to:
  $$\int \sqrt{1-x^{2}} : dx = \int \sqrt{1-\sin^2(t)} \cdot \cos(t)dt = \int \cos^2(t) dt$$

• Integration by Parts:
  To solve $\int \cos^2(t) dt$, use integration by parts with $u = \cos(t)$ and $dv = \cos(t)dt$. This gives $du = -\sin(t)dt$ and $v = \sin(t)$. Applying the formula $\int u dv = uv - \int v du$:
  $$\int \cos^{2}(t) dt = \sin(t)\cos(t) + \int \sin^{2}(t) dt$$

• Final Simplification:
  Next, use the identity $\sin^2(t) = 1 - \cos^2(t)$ to substitute back into the equation, allowing you to solve for $\int \cos^2(t) dt$ directly. Don’t forget to substitute $t = \arcsin(x)$ to convert the result back to terms of $x$.

Example 2: $\int \ln(x^{2} - 2x +5) : dx$ #

This integral combines integration by parts with a simple substitution to simplify the quadratic inside the logarithm.

• Complete the Square:
  First, rewrite the quadratic inside the log by completing the square:
  $x^2 - 2x +5 = (x-1)^2 + 4$. This makes the subsequent substitution step much smoother.

• Integration by Parts:
  Let $u = \ln((x-1)^2 +4)$ (so $du = \frac{2(x-1)}{(x-1)^2 +4} dx$) and $dv = dx$ (so $v = x$). Applying integration by parts:
  $$\int \ln(x^{2} - 2x +5) dx = x\ln((x-1)^2 +4) - \int \frac{2x(x-1)}{(x-1)^2 +4} dx$$

• Variable Substitution:
  Let $u = x-1$, meaning $x = u+1$ and $dx = du$. Substitute into the remaining integral:
  $$\int \frac{2(u+1)u}{u^2 +4} du$$
  Split this into simpler integrals, solve each one, then substitute back $u = x-1$ and combine with the earlier term to get the final result.

内容的提问来源于stack exchange，提问作者Redsbefall