Integration | Jamb Mathematics

Calculation problems involving Integration of explicit algebraic expression

Problem 1: Basic Power Rule

1. Apply the power rule:
   $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$.
2. Integrate term by term:
   $\int 3x^4 \,dx = \frac{3x^5}{5}$,
   $\int -5x^3 \,dx = \frac{-5x^4}{4}$,
   $\int 2x \,dx = \frac{2x^2}{2} = x^2$,
   $\int -7 \,dx = -7x$.
3. Final result:
   $\frac{3x^5}{5} - \frac{5x^4}{4} + x^2 - 7x + C$.

Problem 2: Integral of a Binomial Expansion

1. Integrate each term: $\int x^3 \,dx = \frac{x^4}{4}$,
   $\int 4x^2 \,dx = \frac{4x^3}{3}$,
   $\int -x \,dx = \frac{-x^2}{2}$,
   $\int 6 \,dx = 6x$.
2. Final result:
   $\frac{x^4}{4} + \frac{4x^3}{3} - \frac{x^2}{2} + 6x + C$.

Problem 3: Integral of a Rational Expression

1. Rewrite the expression:
   $\int (6x - 4 + \frac{8}{x}) \,dx$.
2. Integrate term by term: $\int 6x \,dx = \frac{6x^2}{2} = 3x^2$,
   $\int -4 \,dx = -4x$,
   $\int \frac{8}{x} \,dx = 8 \ln |x|$.
3. Final result:
   $3x^2 - 4x + 8 \ln |x| + C$.

Problem 4: Integral of a Fractional Power

1. Apply the power rule: $\int x^{n} \,dx = \frac{x^{n+1}}{n+1}$.
2. Integrate each term: $\int x^{3/2} \,dx = \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2}$,
   $\int -2x^{1/2} \,dx = -2 \times \frac{x^{3/2}}{3/2} = -\frac{4}{3} x^{3/2}$.
3. Final result:
   $\frac{2}{5}x^{5/2} - \frac{4}{3}x^{3/2} + C$.

Problem 5: Integral of a Sum of Roots

1. Rewrite as powers of x: $\int (x^{1/2} + x^{-1/2}) \,dx$.
2. Apply the power rule: $\int x^{1/2} \,dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$,
   $\int x^{-1/2} \,dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$.
3. Final result:
   $\frac{2}{3}x^{3/2} + 2x^{1/2} + C$.

Problem 6: Integration by Substitution

1. Let $u = 2x + 3$, so $du = 2dx$.
2. Rewrite integral:
   $\frac{1}{2} \int u^4 \,du$.
3. Apply power rule:
   $\frac{1}{2} \times \frac{u^5}{5} = \frac{u^5}{10}$.
4. Substitute back $u = 2x + 3$.
5. Final result:
   $\frac{(2x + 3)^5}{10} + C$.

Problem 7: Integral of an Improper Rational Function

1. Simplify:
   $\int (x^2 + x - 1) \,dx$.
2. Integrate each term: $\int x^2 \,dx = \frac{x^3}{3}$,
   $\int x \,dx = \frac{x^2}{2}$,
   $\int -1 \,dx = -x$.
3. Final result:
   $\frac{x^3}{3} + \frac{x^2}{2} - x + C$.

Problem 8: Integral of a Product

1. Expand:
   $\int (x^2 + 2x) \,dx$.
2. Apply power rule: $\int x^2 \,dx = \frac{x^3}{3}$,
   $\int 2x \,dx = x^2$.
3. Final result:
   $\frac{x^3}{3} + x^2 + C$.

Problem 9: Integral of a Quadratic Fraction

1. Let $u = x^2 + 4$, then $du = 2x dx$.
2. Rewrite:
   $\frac{1}{2} \int \frac{du}{u}$.
3. Integrate:
   $\frac{1}{2} \ln |u| + C$.
4. Substitute back $u = x^2 + 4$.
5. Final result:
   $\frac{1}{2} \ln |x^2 + 4| + C$.

Problem 10: Integral of a Polynomial with Negative Exponents

1. Apply the power rule:
   $\int x^n \,dx = \frac{x^{n+1}}{n+1}$, where $n \neq -1$.
2. Integrate each term: $\int x^{-3} \,dx = \frac{x^{-2}}{-2} = -\frac{1}{2}x^{-2}$,
   $\int 5x^{-2} \,dx = \frac{5x^{-1}}{-1} = -5x^{-1}$,
   $\int -2x^{-1} \,dx = -2 \ln |x|$.
3. Final result:
   $-\frac{1}{2}x^{-2} - 5x^{-1} - 2\ln |x| + C$.

Problem 11: Integral of a Cubic Polynomial

1. Apply the power rule to each term: $\int 4x^3 \,dx = \frac{4x^4}{4} = x^4$,
   $\int -6x^2 \,dx = \frac{-6x^3}{3} = -2x^3$,
   $\int 8x \,dx = \frac{8x^2}{2} = 4x^2$,
   $\int -10 \,dx = -10x$.
2. Final result:
   $x^4 - 2x^3 + 4x^2 - 10x + C$.

Problem 12: Integral Using Substitution

1. Let $u = 3x + 5$, so that $du = 3dx \Rightarrow dx = \frac{du}{3}$.
2. Rewrite integral:
   $\int (3x + 5)^2 \,dx = \int u^2 \frac{du}{3} = \frac{1}{3} \int u^2 \,du$.
3. Apply power rule:
   $\frac{1}{3} \times \frac{u^3}{3} = \frac{u^3}{9}$.
4. Substitute back $u = 3x + 5$.
5. Final result:
   $\frac{(3x + 5)^3}{9} + C$.

Problem 13: Integral of a Rational Function with a Linear Denominator

1. Recognize the standard integral:
   $\int \frac{1}{x + a} \,dx = \ln |x + a| + C$.
2. Factor out constant:
   $4 \int \frac{1}{x + 3} \,dx$.
3. Solve:
   $4 \ln |x + 3| + C$.
4. Final result:
   $4\ln |x + 3| + C$.

Problem 14: Integral of a Square Root Expression

1. Let $u = 4x + 1$, so that $du = 4dx$.
2. Rewrite integral:
   $\int \sqrt{u} \frac{du}{4} = \frac{1}{4} \int u^{1/2} \,du$.
3. Apply power rule:
   $\frac{1}{4} \times \frac{u^{3/2}}{3/2} = \frac{u^{3/2}}{6}$.
4. Substitute back $u = 4x + 1$.
5. Final result:
   $\frac{(4x + 1)^{3/2}}{6} + C$.

Problem 15: Integral of a Logarithmic Function

1. Use integration by parts, where:
   $u = \ln x \Rightarrow du = \frac{dx}{x}$,
   $dv = x dx \Rightarrow v = \frac{x^2}{2}$.
2. Apply the integration by parts formula:
   $\int u dv = uv - \int v du$.
3. Compute:
   $\frac{x^2}{2} \ln x - \int \frac{x^2}{2} \times \frac{dx}{x}$.
4. Simplify:
   $\frac{x^2}{2} \ln x - \int \frac{x}{2} dx$.
5. Solve remaining integral:
   $\frac{x^2}{2} \ln x - \frac{x^2}{4} + C$.
6. Final result:
   $\frac{x^2}{2} \ln x - \frac{x^2}{4} + C$.

Calculation problem involving integration of trigonometric function

Problem 1: Basic Sine Integration

1. Recall the basic integral:
   $\int \sin x \,dx = -\cos x + C$.
2. Final Answer:
   $-\cos x + C$.

Problem 2: Basic Cosine Integration

1. Recall the basic integral:
   $\int \cos x \,dx = \sin x + C$.
2. Final Answer:
   $\sin x + C$.

Problem 3: Integral of Tangent

1. Rewrite in terms of sine and cosine:
   $\int \frac{\sin x}{\cos x} \,dx$.
2. Use substitution:
   Let $u = \cos x \Rightarrow du = -\sin x dx$.
3. Rewrite integral:
   $-\int \frac{du}{u} = -\ln |u| + C$.
4. Substitute back $u = \cos x$.
5. Final Answer:
   $-\ln |\cos x| + C$.

Problem 4: Integral of Secant

1. Use the standard formula:
   $\int \sec x \,dx = \ln |\sec x + \tan x| + C$.
2. Final Answer:
   $\ln |\sec x + \tan x| + C$.

Problem 5: Integral of Cosecant

1. Use the standard formula:
   $\int \csc x \,dx = -\ln |\csc x + \cot x| + C$.
2. Final Answer:
   $-\ln |\csc x + \cot x| + C$.

Problem 6: Integral of Cotangent

1. Rewrite in terms of sine and cosine:
   $\int \frac{\cos x}{\sin x} \,dx$.
2. Use substitution:
   Let $u = \sin x \Rightarrow du = \cos x dx$.
3. Rewrite integral:
   $\int \frac{du}{u} = \ln |u| + C$.
4. Substitute back $u = \sin x$.
5. Final Answer:
   $\ln |\sin x| + C$.

Problem 7: Integral of Sin Squared

1. Use identity:
   $\sin^2 x = \frac{1 - \cos 2x}{2}$.
2. Rewrite integral:
   $\int \frac{1 - \cos 2x}{2} \,dx$.
3. Integrate each term separately:
   $\frac{1}{2} \int dx - \frac{1}{2} \int \cos 2x \,dx$.
4. Solve:
   $\frac{x}{2} - \frac{\sin 2x}{4} + C$.
5. Final Answer:
   $\frac{x}{2} - \frac{\sin 2x}{4} + C$.

Problem 8: Integral of Cos Squared

1. Use identity:
   $\cos^2 x = \frac{1 + \cos 2x}{2}$.
2. Rewrite integral:
   $\int \frac{1 + \cos 2x}{2} \,dx$.
3. Integrate each term separately:
   $\frac{x}{2} + \frac{\sin 2x}{4} + C$.
4. Final Answer:
   $\frac{x}{2} + \frac{\sin 2x}{4} + C$.

Problem 9: Integral of Sin and Cos Product

1. Use identity:
   $\sin x \cos x = \frac{1}{2} \sin 2x$.
2. Rewrite integral:
   $\frac{1}{2} \int \sin 2x \,dx$.
3. Integrate:
   $\frac{-\cos 2x}{4} + C$.
4. Final Answer:
   $-\frac{\cos 2x}{4} + C$.

Problem 10: Integral of Sec Squared

1. Recall the standard integral:
   $\int \sec^2 x \,dx = \tan x + C$.
2. Final Answer:
   $\tan x + C$.

Problem 11: Integral of Cosec Squared

1. Recall the standard integral:
   $\int \csc^2 x \,dx = -\cot x + C$.
2. Final Answer:
   $-\cot x + C$.

Problem 12: Integral of Sec x Tan x

1. Recall the standard integral:
   $\int \sec x \tan x \,dx = \sec x + C$.
2. Final Answer:
   $\sec x + C$.

Problem 13: Integral of Cosec x Cot x

1. Recall the standard integral:
   $\int \csc x \cot x \,dx = -\csc x + C$.
2. Final Answer:
   $-\csc x + C$.

Problem 14: Integral of Sine with Shifted Argument

1. Use substitution:
   Let $u = 3x + 5 \Rightarrow du = 3dx$.
2. Rewrite:
   $\frac{1}{3} \int \sin u \,du$.
3. Solve:
   $\frac{-\cos u}{3} + C$.
4. Substitute back $u = 3x + 5$.
5. Final Answer:
   $-\frac{\cos (3x + 5)}{3} + C$.

Problem 15: Integral of Cosine with Shifted Argument

1. Use substitution:
   Let $u = 4x - 2 \Rightarrow du = 4dx$.
2. Rewrite:
   $\frac{1}{4} \int \cos u \,du$.
3. Solve:
   $\frac{\sin u}{4} + C$.
4. Substitute back $u = 4x - 2$.
5. Final Answer:
   $\frac{\sin (4x - 2)}{4} + C$.

Application problem involving area under the curve

Problem 1: Area Under a Parabola

1. The area is given by:
   $A = \int_0^3 x^2 \,dx$.
2. Integrate using the power rule:
   $\int x^2 \,dx = \frac{x^3}{3}$.
3. Evaluate from $0$ to $3$:
   $\left[ \frac{x^3}{3} \right]_0^3 = \frac{3^3}{3} - \frac{0^3}{3} = \frac{27}{3} - 0 = 9$.
4. Final Answer: $9$ square units.

Problem 2: Area Under a Linear Function

1. The area is given by:
   $A = \int_1^5 (4x + 1) \,dx$.
2. Integrate each term separately:
   $\int 4x \,dx = 2x^2$,
   $\int 1 \,dx = x$.
3. Compute:
   $\left[ 2x^2 + x \right]_1^5 = (2(5)^2 + 5) - (2(1)^2 + 1)$.
4. Simplify:
   $(50 + 5) - (2 + 1) = 55 - 3 = 52$.
5. Final Answer: $52$ square units.

Problem 3: Area Between Two Curves

1. Compute the area using:
   $A = \int_{-1}^{2} [(x+2) - x^2] \,dx$.
2. Integrate each term:
   $\int x \,dx = \frac{x^2}{2}$,
   $\int 2 \,dx = 2x$,
   $\int x^2 \,dx = \frac{x^3}{3}$.
3. Compute:
   $\left[ \frac{x^2}{2} + 2x - \frac{x^3}{3} \right]_{-1}^{2}$.
4. Evaluate at $x = 2$ and $x = -1$ and subtract.
5. Final Answer: $\frac{9}{2}$ square units.

Problem 4: Area Under an Absolute Value Function

1. Break into two cases:
   • For $x < 1$, $y = -(x - 1) = -x + 1$.
   • For $x \geq 1$, $y = x - 1$.
2. Compute the areas separately:
   $A_1 = \int_{-1}^{1} (-x + 1) \,dx$,
   $A_2 = \int_{1}^{3} (x - 1) \,dx$.
3. Evaluate and sum.
4. Final Answer: $4$ square units.

Problem 5: Area Under a Trigonometric Curve

1. Compute:
   $A = \int_0^\pi \sin x \,dx$.
2. Integrate:
   $\int \sin x \,dx = -\cos x$.
3. Evaluate:
   $[-\cos x]_0^\pi = (-\cos \pi) - (-\cos 0)$.
4. Simplify:
   $(-(-1)) - (-1) = 1 + 1 = 2$.
5. Final Answer: $2$ square units.

Problem 6: Area Enclosed by a Quadratic and a Line

1. Find points of intersection:
   Solve $x^2 = 2x \Rightarrow x^2 - 2x = 0 \Rightarrow x(x - 2) = 0$.
   So, $x = 0, 2$.
2. Compute area:
   $A = \int_0^2 (2x - x^2) \,dx$.
3. Integrate and evaluate.
4. Final Answer: $\frac{4}{3}$ square units.

Problem 7: Area Under an Exponential Curve

1. Compute:
   $A = \int_0^2 e^x \,dx$.
2. Integrate:
   $\int e^x \,dx = e^x$.
3. Evaluate:
   $[e^x]_0^2 = e^2 - e^0 = e^2 - 1$.
4. Final Answer: $e^2 - 1$ square units.

Problem 8: Area Under a Rational Function

1. Compute:
   $A = \int_1^4 \frac{1}{x} \,dx$.
2. Integrate:
   $\int \frac{1}{x} \,dx = \ln |x|$.
3. Evaluate:
   $[\ln |x|]_1^4 = \ln 4 - \ln 1$.
4. Simplify:
   $\ln 4 - 0 = \ln 4$.
5. Final Answer: $\ln 4$ square units.

Problem 9: Area Between a Sine and a Cosine Curve

1. Compute:
   $A = \int_0^{\pi/4} (\cos x - \sin x) \,dx$.
2. Integrate:
   $\int \cos x \,dx = \sin x$,
   $\int \sin x \,dx = -\cos x$.
3. Evaluate and simplify.
4. Final Answer: $\frac{\sqrt{2}}{2}$ square units.

Problem 10: Area Under a Polynomial Function

1. Compute:
   $A = \int_{-1}^{2} (x^3 - 2x + 1) \,dx$.
2. Integrate term by term:
   • $\int x^3 \,dx = \frac{x^4}{4}$,
   • $\int -2x \,dx = -x^2$,
   • $\int 1 \,dx = x$.
3. Evaluate from $-1$ to $2$: $\left[ \frac{x^4}{4} - x^2 + x \right]_{-1}^{2}$
4. Compute values at $x = 2$ and $x = -1$ and subtract.
5. Final Answer: $\frac{35}{4}$ square units.

Problem 11: Area Under an Exponential Function

1. Compute:
   $A = \int_0^1 3e^{2x} \,dx$.
2. Use substitution:
   Let $u = 2x$, then $du = 2dx$.
3. Solve integral:
   $\int 3e^{2x} \,dx = \frac{3}{2} e^{2x}$.
4. Evaluate:
   $\left[ \frac{3}{2} e^{2x} \right]_0^1$.
5. Compute values:
   $\frac{3}{2} (e^2 - e^0) = \frac{3}{2} (e^2 - 1)$.
6. Final Answer: $\frac{3}{2} (e^2 - 1)$ square units.

Problem 12: Area Enclosed by a Parabola and a Line

1. Find intersection points:
   • Solve $x^2 = 4x$,
   • $x^2 - 4x = 0$,
   • $x(x - 4) = 0$, so $x = 0, 4$.
2. Compute area:
   $A = \int_0^4 (4x - x^2) \,dx$.
3. Integrate:
   • $\int 4x \,dx = 2x^2$,
   • $\int -x^2 \,dx = -\frac{x^3}{3}$.
4. Evaluate from $0$ to $4$.
5. Final Answer: $\frac{32}{3}$ square units.

Problem 13: Area Between Two Trigonometric Curves

1. Compute area:
   $A = \int_0^{\pi/2} (\cos x - \sin x) \,dx$.
2. Integrate:
   • $\int \cos x \,dx = \sin x$,
   • $\int \sin x \,dx = -\cos x$.
3. Evaluate from $0$ to $\frac{\pi}{2}$.
4. Compute values:
   $(\sin \frac{\pi}{2} - (-\cos \frac{\pi}{2})) - (\sin 0 - (-\cos 0))$.
5. Final Answer: $1$ square unit.

Problem 14: Area Enclosed by a Square Root Function

1. Compute:
   $A = \int_1^4 \sqrt{x} \,dx$.
2. Rewrite as exponent:
   $A = \int_1^4 x^{1/2} \,dx$.
3. Integrate using the power rule:
   $\int x^{n} \,dx = \frac{x^{n+1}}{n+1}$.
4. Compute:
   • $\int x^{1/2} \,dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$.
5. Evaluate: $\left[ \frac{2}{3} x^{3/2} \right]_1^4$.
6. Compute values at $x = 4$ and $x = 1$, then subtract.
7. Final Answer: $\frac{14}{3}$ square units.

Problem 15: Area Under a Rational Function

1. Compute:
   $A = \int_1^3 \frac{2}{x^2} \,dx$.
2. Rewrite as exponent:
   $A = \int_1^3 2x^{-2} \,dx$.
3. Use power rule:
   $\int x^n \,dx = \frac{x^{n+1}}{n+1}$.
4. Compute:
   • $\int 2x^{-2} \,dx = 2 \times \frac{x^{-1}}{-1} = -\frac{2}{x}$.
5. Evaluate: $\left[ -\frac{2}{x} \right]_1^3$.
6. Compute values at $x = 3$ and $x = 1$, then subtract.
7. Final Answer: $\frac{4}{3}$ square units.

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