Tutorial Week 12
This week’s tutorial will be your last tutorial for this course.
Integral Theorems
Q1: Compute \(\int^3_3 \frac{\cos(e^{x})}{x^{2}+e^{x}} \, dx\).
Q2: Compute \(\int^{-\frac{\pi}{3}}_{\frac{\pi}{3}} \frac{x}{\cos\left(x\right)} \, dx\).
Q3: If \(\int_{1}^{5} f(x) + 4 \, dx = 8\), \(\int_1^2 f(x) + x \, dx = 2\), find \(\int_2^5 f(x) \, dx\).
Fundamental Theorem Of Calculus I
Q4: Given \(f(x) = \int_{\sqrt{x}}^{3x} 5t + 3 \, dt\), find \(\frac{df}{dx}\).
Q5: Differentiate \(\int_{-8x}^{3^{2x}} 2t^3 \, dt\) in terms of \(x\).
Q6: Differentiate \(\int_{-5}^{sin(x) + cos(x)} ln(t+10) + tan^{-1}(t) \, dt\) in terms of \(x\).
Subtitution Method (aka u-substitution)
Q7: Compute \(\int x\left(x^{2}-1\right)^{6} \, dx\).
Q8: Compute \(\int \frac{2x+5}{x^{2}+5x} \, dx\).
Q9: Compute \(\int \frac{\sec^{2}\left(x^{-2}\right)}{5x^{3}} \, dx\).
Q10: Compute \(\int \cos^{3}\left(x\right)\ \sqrt{\sin\left(x\right)} \, dx\).
Q11: Compute \(\int x\sec^{2}\left(x^{2}\right)\tan^{4}\left(x^{2}\right) \, dx\).
