Indefinite Integrals and Substitution Method

Indefinite Integrals and Substitution Method

• The indefinite integral of a function $f(x)$, denoted by $\int f(x) dx$, is a family of functions whose derivative is $f(x)$.
• The general form of an indefinite integral is $F(x)+C$, where $F'(x) = f(x)$ and C is an arbitrary constant of integration.
• The substitution method is a technique used to solve indefinite integrals of the form $\int f(g(x)).g'(x)dx$.
• The method involves substituting $u = g(x)$ and $du = g'(x) dx$, transforming the integral into $\int f(u) du$.

Integration by Substitution

• The chain rule in differentiation states that $\frac{d}{dx}(F(g(x))) = F'(g(x)).g'(x)$.
• The substitution method essentially reverses the chain rule, allowing us to integrate functions involving composite functions.

Steps for Substitution Method

• Step 1: Identify a suitable substitution $u = g(x)$.
• Step 2: Find the derivative of $u$ with respect to $x$ and solve for $dx$.
• Step 3: Substitute $u$ and $dx$ in the integral, transforming it into an integral involving only $u$.
• Step 4: Integrate the transformed integral with respect to $u$.
• Step 5: Replace $u$ with the original expression $g(x)$.

Examples of Substitution Method

• Example 1: $\int (x^{2} + x)(3x + 1)dx$

  • Substituting $u = x^{2} + x$ and $du = (3x + 1) dx$ simplifies the integral to $\int u du$.
  • The result is $\frac{1}{6}(x^{2} + x)^{6} + C$.

• Example 2: $\int \sqrt{2x + 1} dx$

  • Substitute $u = 2x + 1$ and $du = 2 dx$.
  • The resulting integral $\frac{1}{2} \int u^{\frac{1}{2}} du$ evaluates to $(\frac{1}{3}(2x + 1)^{\frac{3}{2}} + C$.

• Example 3: $\int sec^{2}(5x + 1). 5dx$

  • Use the substitution $u = 5x + 1$ and $du = 5 dx$.
  • Integrate $\int sec^{2}udu$ to obtain $tanu + C$.
  • Finally, replace $u$ with $5x + 1$ to get $tan(5x + 1) + C$.

• Example 4: $\int cos(7\theta + 3) d\theta $

  • Substitute $u = 7\theta + 3$ and $du = 7d\theta$.
  • This leads to $\frac{1}{7} \int cos u du$, which evaluates to $\frac{1}{7} sin (7\theta + 3) + C$.

• Example 5: $\int x^{2} cos x^{3} dx$

  • Substitute $u = x^{3}$ and $du = 3x^{2} dx$.
  • The integral becomes $\frac{1}{3} \int cos u du$, which evaluates to $\frac{1}{3} sin x^{3} + C$.

• Example 6: $\int x \sqrt{2x + 1} dx$

  • Substitute $u = 2x + 1$, $x = \frac{u-1}{2}$, and $du = 2 dx$.
  • Integration leads to $\frac{1}{10}(2x + 1)^{\frac{5}{2}} - \frac{1}{6}(2x + 1)^{\frac{3}{2}} + C$.

• Example 7:

  • a) $\int sin^{2}x dx = \frac{1}{2}x - \frac{1}{4} sin 2x + C$
  • b) $\int cos^{2}x dx = \frac{1}{2}x + \frac{1}{4} sin 2x + C$
  • c) $\int (1 - 2sinx) sin^{2}xdx = \frac{-1}{8} cos(4x) + C$

• Example 8: $\int \frac{2z}{\sqrt{z^{2} + 1}} dz$

  • Substitute $u = z^{2} + 1$ and $du = 2z dz$.
  • The integral becomes $\int u^{\frac{-1}{2}} du$, which evaluates to $2 \sqrt{(z^{2} + 1)} + C$.

Exercises

• 1/245: $\int 2(2x + 4)^{5} dx = \frac{(2x + 4)^{6}}{6} + C$
• 3/245: $\int 2x (x + 5)^{4} dx = \frac{(x + 5)^{5}}{5} + C$
• 5/245: $\int (3x + 2)(3x^{2} + 4x)^{4} dx = \frac{1}{10} (3x^{2} + 4x)^{5} + C$
• 7/245: $\int sin 3x dx = \frac{-1}{3} cos 3x + C$
• 9/245: $\int sec(2t) tan(2t) dt = \frac{1}{2} sec 2t + C$
• 11/245: $\int \frac{9x^{2}}{\sqrt{1 - x^{3}}} dx = -6\sqrt{1 - x^{3}} +C$
• 17/245: $\int \sqrt{3 - 2S} dS = \frac{-1}{3}(3-2S)^{\frac{3}{2}} + C$
• 19/245: $\int \theta \sqrt[4]{1 - \theta^{2}} d\theta = \frac{-1}{2} \int (1 - \theta^{2})^{\frac{1}{4}}.