Welcome back, class! We have just wrapped up a challenging chapter on advanced integration techniques. I know Chapter 7 can be a hurdle, as it requires you to synthesize almost everything you've learned about Calculus so far. To help you study and review, I have uploaded the full Chapter 7 Test Solutions.

Below, I want to highlight a few specific problems from the test that illustrate key strategies you need to master.

1. Integration by Parts

The first problem on the test, $\int_1^4 \sqrt{y} \ln(y) \, dy$, is a classic candidate for Integration by Parts. Remember the formula:

$$ \int u \, dv = uv - \int v \, du $$

The key here is choosing the right $u$. Using the acronym LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential), we see that Logarithmic functions come first. Therefore, we set $u = \ln(y)$ and $dv = \sqrt{y} \, dy$. This transforms a difficult integral into a manageable power rule problem.

2. Handling Trigonometric Powers

Problem 2 asks you to evaluate $\int \sin^5(t) \cos^4(t) \, dt$. When you see an odd power of sine or cosine, the strategy is to "peel off" one factor to save for your $du$, and convert the rest using the identity $\sin^2(t) + \cos^2(t) = 1$.

  • Save one $\sin(t)$ for the differential.
  • Convert $\sin^4(t)$ into $(1-\cos^2(t))^2$.
  • Use $u$-substitution where $u = \cos(t)$.

3. The "Constant" Trap

Take a look at Problem 3: $\int 3e^6 \, dx$. Did this one make you pause? It is very common to see an exponential function and immediately try to integrate it as such. However, notice that $3e^6$ contains no variable. It is simply a constant number! The integral is just the constant times $x$:

$$ \int 3e^6 \, dx = 3e^6 x + C $$

Always check your variable of integration before starting!

4. Partial Fractions and Long Division

Problem 7 involving $\int \frac{5x^2-6}{x^2-4x-12} \, dx$ demonstrates a crucial first step in Partial Fraction Decomposition. Before you can separate the fractions, you must compare the degrees of the numerator and denominator.

Since the degree of the numerator (2) is equal to the degree of the denominator (2), you must perform polynomial long division first. Only after dividing can you apply partial fractions to the remainder.

5. Algebraic Manipulation

Finally, the last example in the notes shows a rationalizing substitution for $\int \sqrt{\frac{7-x}{7+x}} \, dx$. While a $u$-substitution works, the solution shows a clever algebraic trick: multiplying the numerator and denominator by $\sqrt{7-x}$. This clears the square root from the numerator, allowing you to split the integral into two parts: a standard arc-sine integral and a simple $u$-substitution.

Please download the attached PDF to see the full, step-by-step working for every problem. Keep practicing these techniques, and don't hesitate to reach out if you have questions!