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Mathematics LibreTexts

8.E: Techniques of Integration (Exercises)

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    These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

    8.1: Substitution

    Find the antiderivatives or evaluate the definite integral in each problem.

    Ex 8.1.1 \(\int (1-t)^9\,dt\) (answer)

    Ex 8.1.2 \(\int (x^2+1)^2\,dx\) (answer)

    Ex 8.1.3 \(\int x(x^2+1)^{100}\,dx\) (answer)

    Ex 8.1.4 \(\int {1\over\root 3 \of {1-5t}}\,dt\) (answer)

    Ex 8.1.5 \(\int \sin^3x\cos x\,dx\) (answer)

    Ex 8.1.6 \(\int x\sqrt{100-x^2}\,dx\) (answer)

    Ex 8.1.7 \(\int {x^2\over\sqrt{1-x^3}}\,dx\) (answer)

    Ex 8.1.8 \(\int \cos(\pi t)\cos\bigl(\sin(\pi t)\bigr)\,dt\) (answer)

    Ex 8.1.9 \(\int {\sin x\over\cos^3 x}\,dx\) (answer)

    Ex 8.1.10 \(\int\tan x\,dx\) (answer)

    Ex 8.1.11 \(\int_0^\pi\sin^5(3x)\cos(3x)\,dx\) (answer)

    Ex 8.1.12 \(\int\sec^2x\tan x\,dx\) (answer)

    Ex 8.1.13 \(\int_0^{\sqrt{\pi}/2} x\sec^2(x^2)\tan(x^2)\,dx\) (answer)

    Ex 8.1.14 \(\int {\sin(\tan x)\over\cos^2x}\,dx\) (answer)

    Ex 8.1.15 \(\int_3^4 {1\over(3x-7)^2}\,dx\) (answer)

    Ex 8.1.16 \(\int_0^{\pi/6}(\cos^2x - \sin^2x)\,dx\) (answer)

    Ex 8.1.17 \(\int {6x\over(x^2 - 7)^{1/9}}\,dx\) (answer)

    Ex 8.1.18 \(\int_{-1}^1 (2x^3-1)(x^4-2x)^6\,dx\) (answer)

    Ex 8.1.19 \(\int_{-1}^1 \sin^7 x\,dx\) (answer)

    Ex 8.1.20 \(\int f(x) f'(x)\,dx\) (answer)

    8.2: Powers of sine and cosine

    Find the antiderivatives.

    Ex 8.2.1 \( \int \sin^2 x\,dx\) (answer)

    Ex 8.2.2 \( \int \sin^3 x\,dx\) (answer)

    Ex 8.2.3 \( \int \sin^4 x\,dx\) (answer)

    Ex 8.2.4 \( \int \cos^2 x\sin^3 x\,dx\) (answer)

    Ex 8.2.5 \( \int \cos^3 x\,dx\) (answer)

    Ex 8.2.6 \( \int \sin^2 x\cos^2 x\,dx\) (answer)

    Ex 8.2.7 \( \int \cos^3 x \sin^2 x\,dx\) (answer)

    Ex 8.2.8 \( \int \sin x (\cos x)^{3/2}\,dx\) (answer)

    Ex 8.2.9 \( \int \sec^2 x\csc^2 x\,dx\) (answer)

    Ex 8.2.10 \( \int \tan^3x \sec x\,dx\) (answer)

    8.3: Trigonometric Substitutions

    Find the antiderivatives.

    Ex 8.3.1 $\ds\int\csc x\,dx$ (answer)

    Ex 8.3.2 $\ds\int\csc^3 x\,dx$ (answer)

    Ex 8.3.3 $\ds\int\sqrt{x^2-1}\,dx$ (answer)

    Ex 8.3.4 $\ds\int\sqrt{9+4x^2}\,dx$ (answer)

    Ex 8.3.5 $\ds\int x\sqrt{1-x^2}\,dx$ (answer)

    Ex 8.3.6 $\ds\int x^2\sqrt{1-x^2}\,dx$ (answer)

    Ex 8.3.7 $\ds\int{1\over\sqrt{1+x^2}}\,dx$ (answer)

    Ex 8.3.8 $\ds\int\sqrt{x^2+2x}\,dx$ (answer)

    Ex 8.3.9 $\ds\int{1\over x^2(1+x^2)}\,dx$ (answer)

    Ex 8.3.10 $\ds\int{x^2\over\sqrt{4-x^2}}\,dx$ (answer)

    Ex 8.3.11 $\ds\int{\sqrt{x}\over\sqrt{1-x}}\,dx$ (answer)

    Ex 8.3.12 $\ds\int{x^3\over\sqrt{4x^2-1}}\,dx$ (answer)

    8.4: Integration by Parts

    Find the antiderivatives.

    Ex 8.4.1 \( \int x\cos x\,dx\) (answer)

    Ex 8.4.2 \( \int x^2\cos x\,dx\) (answer)

    Ex 8.4.3 \( \int xe^x\,dx\) (answer)

    Ex 8.4.4 \( \int xe^{x^2}\,dx\) (answer)

    Ex 8.4.5 \( \int \sin^2 x\,dx\) (answer)

    Ex 8.4.6 \( \int \ln x\,dx\) (answer)

    Ex 8.4.7 \( \int x\arctan x\,dx\) (answer)

    Ex 8.4.8 \( \int x^3\sin x\,dx\) (answer)

    Ex 8.4.9 \( \int x^3\cos x\,dx\) (answer)

    Ex 8.4.10 \( \int x\sin^2 x\,dx\) (answer)

    Ex 8.4.11 \( \int x\sin x\cos x\,dx\) (answer)

    Ex 8.4.12 \( \int \arctan(\sqrt x)\,dx\) (answer)

    Ex 8.4.13 \( \int \sin(\sqrt x)\,dx\) (answer)

    Ex 8.4.14 \( \int\sec^2 x\csc^2 x\,dx\) (answer)

    8.5: Rational Functions

    Find the antiderivatives.

    Ex 8.5.1 \( \int {1\over 4-x^2}\,dx\) (answer)

    Ex 8.5.2 \( \int {x^4\over 4-x^2}\,dx\) (answer)

    Ex 8.5.3 \( \int {1\over x^2+10x+25}\,dx\) (answer)

    Ex 8.5.4 \( \int {x^2\over 4-x^2}\,dx\) (answer)

    Ex 8.5.5 \( \int {x^4\over 4+x^2}\,dx\) (answer)

    Ex 8.5.6 \( \int {1\over x^2+10x+29}\,dx\) (answer)

    Ex 8.5.7 \( \int {x^3\over 4+x^2}\,dx\) (answer)

    Ex 8.5.8 \( \int {1\over x^2+10x+21}\,dx\) (answer)

    Ex 8.5.9 \( \int {1\over 2x^2-x-3}\,dx\) (answer)

    Ex 8.5.10 \( \int {1\over x^2+3x}\,dx\) (answer)

    8.6: Numerical Integration

    In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.) If you have access to Sage or similar software, approximate each integral to two decimal places. You can use this Sage worksheet to get started.

    Ex 8.6.1 \( \int_1^3 x\,dx\) (answer)

    Ex 8.6.2 \( \int_0^3 x^2\,dx\) (answer)

    Ex 8.6.3 \( \int_2^4 x^3\,dx\) (answer)

    Ex 8.6.4 \( \int_1^3 {1\over x}\,dx\) (answer)

    Ex 8.6.5 \( \int_1^2 {1\over 1+x^2}\,dx\) (answer)

    Ex 8.6.6 \( \int_0^1 x\sqrt{1+x}\,dx\) (answer)

    Ex 8.6.7 \( \int_1^5 {x\over 1+x}\,dx\) (answer)

    Ex 8.6.8 \( \int_0^1 \sqrt{x^3+1}\,dx\) (answer)

    Ex 8.6.9 \( \int_0^1 \sqrt{x^4+1}\,dx\) (answer)

    Ex 8.6.10 \( \int_1^4 \sqrt{1+1/x}\,dx\) (answer)

    Ex 8.6.11 Using Simpson's rule on a parabola \(f(x)\), even with just two subintervals, gives the exact value of the integral, because the parabolas used to approximate \(f\) will be \(f\) itself. Remarkably, Simpson's rule also computes the integral of a cubic function \(f(x)=ax^3+bx^2+cx+d\) exactly. Show this is true by showing that

    \[ \int_{x_0}^{x_2} f(x)\,dx={x_2-x_0\over3\cdot2}(f(x_0)+4f((x_0+x_2)/2)+f(x_2)). \]

    This does require a bit of messy algebra, so you may prefer to use Sage.

    8.7: Additional exercises

    These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way.

    Ex 8.7.1 \(\int (t+4)^3\,dt\) (answer)

    Ex 8.7.2 \(\int t(t^2-9)^{3/2}\,dt\) (answer)

    Ex 8.7.3 \(\int (e^{t^2}+16)te^{t^2}\,dt\) (answer)

    Ex 8.7.4 \(\int \sin t\cos 2t\,dt\) (answer)

    Ex 8.7.5 \(\int \tan t\sec^2t\,dt\) (answer)

    Ex 8.7.6 \(\int {2t+1\over t^2+t+3}\,dt\) (answer)

    Ex 8.7.7 \(\int {1\over t(t^2-4)}\,dt\) (answer)

    Ex 8.7.8 \(\int {1\over (25-t^2)^{3/2}}\,dt\) (answer)

    Ex 8.7.9 \(\int {\cos 3t\over\sqrt{\sin3t}}\,dt\) (answer)

    Ex 8.7.10 \(\int t\sec^2 t\,dt\) (answer)

    Ex 8.7.11 \(\int {e^t\over \sqrt{e^t+1}}\,dt\) (answer)

    Ex 8.7.12 \(\int \cos^4 t\,dt\) (answer)

    Ex 8.7.13 \(\int {1\over t^2+3t}\,dt\) (answer)

    Ex 8.7.14 \(\int {1\over t^2\sqrt{1+t^2}}\,dt\) (answer)

    Ex 8.7.15 \(\int {\sec^2t\over (1+\tan t)^3}\,dt\) (answer)

    Ex 8.7.16 \(\int t^3\sqrt{t^2+1}\,dt\) (answer)

    Ex 8.7.17 \(\int e^t\sin t\,dt\) (answer)

    Ex 8.7.18 \(\int (t^{3/2}+47)^3\sqrt{t}\,dt\) (answer)

    Ex 8.7.19 \(\int {t^3\over (2-t^2)^{5/2}}\,dt\) (answer)

    Ex 8.7.20 \(\int {1\over t(9+4t^2)}\,dt\) (answer)

    Ex 8.7.21 \(\int {\arctan 2t\over 1+4t^2}\,dt\) (answer)

    Ex 8.7.22 \(\int {t\over t^2+2t-3}\,dt\) (answer)

    Ex 8.7.23 \(\int \sin^3 t\cos^4 t\,dt\) (answer)

    Ex 8.7.24 \(\int {1\over t^2-6t+9}\,dt\) (answr)

    Ex 8.7.25 \(\int {1\over t(\ln t)^2}\,dt\) (answer)

    Ex 8.7.26 \(\int t(\ln t)^2\,dt\) (answer)

    Ex 8.7.27 \(\int t^3e^{t}\,dt\) (answer)

    Ex 8.7.28 \(\int {t+1\over t^2+t-1}\,dt\) (answer)