Session 04 - U Substitution

We want to evaluate the following indefinite integral:

This problem is a perfect candidate for $u$-substitution because the derivative of the expression inside the radical, $x^2 + x$, is exactly the other term in the integrand, $2x + 1$.

Step 1: Choose $u$ and find $du$

Let's define our substitute variable $u$ as the inside function:

Now, take the derivative of $u$ with respect to $x$ to find $du$:

Step 2: Substitute $u$ and $du$ into the original integral

Notice how the pieces of our original integral perfectly match our substitution components:

• $\sqrt[5]{x^2+x}$ becomes $\sqrt[5]{u}$ (or $u^{1/5}$)
• $(2x + 1) \, dx$ becomes $du$

Rewriting the integral gives:

Step 3: Integrate with respect to $u$

Using the power rule for integration, $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$:

Step 4: Substitute the original expression back in

Finally, replace $u$ with our original expression $x^2 + x$ to get the final answer in terms of $x$:

We want to evaluate the following indefinite integral:

This is another classic $u$-substitution problem. The degree of the polynomial in the denominator ($x^2+1$) is 2, and the degree of the numerator ($x$) is 1. Since the numerator is a scalar multiple of the derivative of the denominator, $u$-substitution is the perfect tool.

Step 1: Choose $u$ and find $du$

Let's set $u$ equal to the entire denominator:

Now, take the derivative of $u$ with respect to $x$:

Step 2: Adjust for the coefficients

Looking at our original integral, the numerator only has $x \, dx$, not $2x \, dx$. We can adjust our $du$ equation by dividing both sides by 2:

Step 3: Substitute into the original integral

Now replace the terms in the original integral with $u$ and $du$:

• The denominator $x^2 + 1$ becomes $u$.
• The numerator and differential $x \, dx$ become $\frac{1}{2} \, du$.

Pull the constant factor $\frac{1}{2}$ outside the integral:

Step 4: Integrate with respect to $u$

The integral of $\frac{1}{u}$ is a standard rule: $\int \frac{1}{u} \, du = \ln|u| + C$.

Step 5: Substitute the original expression back in

Replace $u$ with $x^2 + 1$:

We want to evaluate the following definite integral:

This problem is a classic trigonometry-based $u$-substitution. Since the derivative of $\tan(x)$ is exactly $\sec^2(x)$, we have a perfect pairing ready to go. Because this is a definite integral (it has upper and lower limits), we also need to update our limits of integration when we switch from $x$ to $u$.

Step 1: Choose $u$ and find $du$

Let's choose our substitution variable:

Now, take the derivative of $u$ with respect to $x$:

Step 2: Convert the limits of integration

Since we are changing variables from $x$ to $u$, we must plug the original $x$ limits into our substitution equation ($u = \tan(x)$) to find the new $u$ limits.

• Lower limit ($x = 0$):
  $$u = \tan(0) = 0$$
• Upper limit ($x = \frac{\pi}{4}$):
  $$u = \tan\left(\frac{\pi}{4}\right) = 1$$

Step 3: Substitute into the integral

Now, swap out all the $x$ components for their $u$ equivalents:

• The limits change from $0 \to \frac{\pi}{4}$ to $0 \to 1$.
• $\tan(x)$ becomes $u$.
• $\sec^2(x) \, dx$ becomes $du$.

Step 4: Integrate and evaluate

Using the power rule for integration ($\int u \, du = \frac{1}{2}u^2$), we evaluate from $0$ to $1$:

Now, apply the Fundamental Theorem of Calculus:

We want to evaluate the following definite integral:

This problem is a great candidate for $u$-substitution. If we rewrite the fraction slightly, it looks like $\int_{1}^{3} e^{1/x} \cdot \frac{1}{x^2} \, dx$. Since the derivative of $\frac{1}{x}$ is closely related to $\frac{1}{x^2}$, we can set up our substitution perfectly.

Step 1: Choose $u$ and find $du$

Let's choose the exponent of $e$ as our substitute variable:

Now, take the derivative of $u$ with respect to $x$ using the power rule:

Since our original integral has a positive $\frac{1}{x^2} \, dx$, let's move the negative sign to the other side:

Step 2: Convert the limits of integration

Because this is a definite integral, we need to update our lower and upper limits from $x$-values to $u$-values using our formula $u = \frac{1}{x}$.

• Lower limit ($x = 1$):
  $$u = \frac{1}{1} = 1$$
• Upper limit ($x = 3$):
  $$u = \frac{1}{3}$$

Step 3: Substitute into the integral

Now, swap the $x$ variables and limits with their $u$ counterparts:

• The limits change from $1 \to 3$ to $1 \to \frac{1}{3}$.
• $e^{1/x}$ becomes $e^u$.
• $\frac{1}{x^2} \, dx$ becomes $-du$.

Pro-Tip: You can use the negative sign outside the integral to flip the lower and upper limits of integration back into standard increasing order:

Step 4: Integrate and evaluate

The integral of $e^u$ is simply $e^u$. Now we evaluate it from $\frac{1}{3}$ to $1$:

Apply the Fundamental Theorem of Calculus:

• Step 1: Choose $u$ and calculate $du$ . The inner function of the sine expression is $2x$.

  $$u = 2x$$
  $$du = 2 \, dx$$

• Step 2: Balance the constant factor. The original integrand contains only $dx$, not $2 \, dx$. Divide both sides by 2 to balance the differential component:

  $$\frac{1}{2} \, du = dx$$

• Step 3: Transform the integration limits . Use the boundary tracking formula $u = 2x$ to convert the limits from $x$ to $u$:

  • Lower limit: When $x = 0 \implies u = 2(0) = 0$
  • Upper limit: When $x = \frac{\pi}{2} \implies u = 2\left(\frac{\pi}{2}\right) = \pi$

• Step 4: Rewrite and integrate in terms of $u$ . Substitute the updated limits ($0 \to \pi$), replace $2x$ with $u$, and swap $dx$ for $\frac{1}{2} \, du$:

  $$\int_{0}^{\pi} \sin(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{\pi} \sin(u) \, du$$

  The antiderivative of $\sin(u)$ is $-\cos(u)$:

  $$\left[ -\frac{1}{2}\cos(u) \right]_{0}^{\pi}$$

• Step 5: Numerically evaluate the boundaries . Apply the fundamental integration limits directly to the $u$ terms without reverting back to $x$:

  $$\left(-\frac{1}{2}\cos(\pi)\right) - \left(-\frac{1}{2}\cos(0)\right)$$

  Knowing $\cos(\pi) = -1$ and $\cos(0) = 1$:

  $$\left(-\frac{1}{2} \cdot (-1)\right) + \left(\frac{1}{2} \cdot 1\right) = \frac{1}{2} + \frac{1}{2} = 1$$

We want to evaluate the following indefinite integral:

This problem uses a special twist on $u$-substitution often called back-substitution. Notice that if we set $u = 3-x$, its derivative is just $-1$, which doesn't automatically cancel out the $x$ sitting outside the radical. To fix this, we will solve our substitution equation for $x$ and substitute that in too!

Step 1: Choose $u$ and find $du$

Let's choose the expression under the radical to be our $u$:

Now, find $du$:

Step 2: Solve for $x$ in terms of $u$

Since we have an extra $x$ left over in the integrand, let's rearrange our original $u$ equation to isolate $x$:

Step 3: Substitute everything into the original integral

Now replace every single part of the original $x$-integral with its $u$ equivalent:

• $x$ becomes $(3 - u)$
• $\sqrt{3-x}$ becomes $\sqrt{u}$ (or $u^{1/2}$)
• $dx$ becomes $-du$

Pull the negative sign out front:

Step 4: Distribute and integrate

Before integrating, distribute $u^{1/2}$ into the parentheses:

Distribute the negative sign to make the power rule cleaner:

Now, apply the power rule to each term:

• For $u^{3/2}$: $\frac{2}{5}u^{5/2}$
• For $3u^{1/2}$: $2u^{3/2}$

Putting it together:

Step 5: Substitute the original expression back in

Finally, replace $u$ with $3-x$:

We want to evaluate the following definite integral:

This problem is a classic $u$-substitution involving trigonometric functions. The inside function is $x^2$, and its derivative is closely related to the $4x$ term sitting outside the cosine function. Because it is a definite integral, we will also update our limits of integration as we switch from $x$ to $u$.

Step 1: Choose $u$ and find $du$

Let's choose the inside function of the cosine term to be our $u$:

Now, take the derivative of $u$ with respect to $x$:

Step 2: Adjust for the coefficients

Our original integral contains $4x \, dx$, but our derivative gives us $2x \, dx$. To match the integral perfectly, we can multiply both sides of our $du$ equation by 2:

Step 3: Convert the limits of integration

Since we are switching variables from $x$ to $u$, we must transform the upper and lower boundaries using our substitution equation $u = x^2$.

• Lower limit ($x = 0$):
  $$u = (0)^2 = 0$$
• Upper limit ($x = \sqrt{\pi}$):
  $$u = (\sqrt{\pi})^2 = \pi$$

Step 4: Substitute into the integral

Now, replace all components of the original integral with our new $u$ terms:

• The limits change from $0 \to \sqrt{\pi}$ to $0 \to \pi$.
• $\cos(x^2)$ becomes $\cos(u)$.
• $4x \, dx$ becomes $2 \, du$.

Pull the constant factor 2 outside the integral:

Step 5: Integrate and evaluate

The antiderivative of $\cos(u)$ is $\sin(u)$. Now we evaluate this from $0$ to $\pi$:

Apply the Fundamental Theorem of Calculus: