Techniques of Integration: math practice, College – download the PDF

1 Parts with cosine ★★★

Find \( \int x\cos x\,dx \).

2 Parts with sin 2x ★★★

Find \( \int x\sin 2x\,dx \).

3 Odd power of cosine ★★★

Find \( \int\cos^3x\,dx \).

4 Sine times a fourth power ★★★

Find \( \int\sin x\cos^4x\,dx \).

5 Simple partial fractions ★★★

Find \( \int\dfrac{dx}{x^2-4} \).

6 A first improper integral ★★★

Evaluate \( \int_1^{\infty}x^{-3}dx \).

7 Pick the technique ★★★

Name the most efficient method for each integral (do not compute them).

1. \( \int xe^{x^2}dx \)
2. \( \int xe^{3x}dx \)
3. \( \int\dfrac{dx}{x^2-9} \)
4. \( \int\sin^2x\,dx \)

8 Double parts ★★★

Evaluate \( \int_0^1x^2e^x\,dx \).

9 Root times logarithm ★★★

Evaluate \( \int_1^4\sqrt x\,\ln x\,dx \).

10 Even powers ★★★

Find \( \int\sin^2x\cos^2x\,dx \).

11 Area of a quarter disk ★★★

Use a trigonometric substitution to evaluate \( \int_0^2\sqrt{4-x^2}\,dx \), then explain the answer geometrically.

12 Two linear factors ★★★

Find \( \int\dfrac{2x+3}{x^2+x-2}dx \).

13 An exponential tail ★★★

Evaluate \( \int_0^{\infty}xe^{-3x}dx \).

14 Integral of ln x near zero ★★★

Evaluate \( \int_0^1\ln x\,dx \). The integrand is unbounded at 0.

15 Drug exposure ★★★

After an injection, a drug’s concentration in the blood is \( C(t)=8te^{-0.5t} \) mg/L, where \(t\) is in hours. The total exposure is \( \int_0^{\infty}C(t)\,dt \) (in mg·h/L). Compute it.

16 A cycle of exponential and cosine ★★★

Find \( \int e^{2x}\cos x\,dx \).

17 Tangent substitution with a square ★★★

Find \( \int\dfrac{dx}{x^2\sqrt{x^2+9}} \).

18 Substitution then parts ★★★

Find \( \int x^3e^{x^2}dx \).

19 Partial fractions and a limit ★★★

Show that \( \int_1^{\infty}\dfrac{dx}{x(x^2+1)} \) converges and find its value.

20 Estimating pi ★★★

Since \( \int_0^1\dfrac4{1+x^2}dx=\pi \), use \(n=4\) to compute \(T_4\) and \(S_4\) for this integral. Compare each with \(\pi\approx3.14159\).

21 Telescoping improper integral ★★★

Evaluate \( \int_1^{\infty}\dfrac{dx}{x(x+1)} \), and use the result to explain why \( \int_1^{\infty}\dfrac{dx}{x^2+x+1} \) converges.

22 A slow divergence ★★★

Show that \( \int_2^{\infty}\dfrac{dx}{x\ln x} \) diverges, even though the integrand tends to 0. Use a substitution.