JEE MATHS PRACTICE - MAINS TO ADVANCED - CALCULUS - DEFINITE AND INDEFINITE INTEGRALS

 I will now provide 25 concept-based solved examples on Definite and Indefinite Integrals, arranged from easy to challenging levels.

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PART 1: Basic Problems (Easy Level)

1. Evaluate $\int x^2 \, dx$.

Solution:
Using the power rule:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$$ $$\int x^2 \, dx = \frac{x^3}{3} + C.$$

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2. Evaluate $\int (3x^2 - 5x + 2) \, dx$.

Solution:

$$\int 3x^2 \, dx = x^3, \quad \int -5x \, dx = -\frac{5x^2}{2}, \quad \int 2 \, dx = 2x.$$ $$\Rightarrow x^3 - \frac{5x^2}{2} + 2x + C.$$

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3. Evaluate $\int e^x \, dx$.

Solution:

$$\int e^x \, dx = e^x + C.$$

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4. Evaluate $\int \cos x \, dx$.

Solution:

$$\int \cos x \, dx = \sin x + C.$$

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5. Evaluate $\int \frac{1}{x} \, dx$.

Solution:

$$\int \frac{1}{x} \, dx = \ln |x| + C.$$

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PART 2: Intermediate Level Problems

6. Evaluate $\int x e^x \, dx$.

Solution (By Integration by Parts):
Let $u = x$, so $du = dx$. Let $dv = e^x dx$, so $v = e^x$.

$$\int x e^x \, dx = x e^x - \int e^x dx = x e^x - e^x + C.$$

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7. Evaluate $\int \ln x \, dx$.

Solution:
Using integration by parts:
Let $u = \ln x$, so $du = \frac{dx}{x}$, and let $dv = dx$, so $v = x$.

$$\int \ln x \, dx = x \ln x - x + C.$$

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8. Evaluate $\int \tan x \, dx$.

Solution:

$$\int \tan x \, dx = \ln |\sec x| + C.$$

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9. Evaluate $\int \frac{dx}{x^2 + a^2}$.

Solution:

$$\frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C.$$

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10. Evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$.

Solution:

$$\sin^{-1} \left( \frac{x}{a} \right) + C.$$

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PART 3: Definite Integrals and Applications

11. Evaluate $\int_0^1 (x^2 + 2x + 1) dx$.

Solution:

$$\int (x^2 + 2x + 1) \, dx = \frac{x^3}{3} + x^2 + x.$$

Evaluating from $0$ to $1$:

$$\left( \frac{1}{3} + 1 + 1 \right) - (0) = \frac{7}{3}.$$

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12. Evaluate $\int_0^\pi \sin x \, dx$.

Solution:

$$\int \sin x \, dx = -\cos x.$$

Evaluating from $0$ to $\pi$:

$$(-\cos \pi) - (-\cos 0) = (1 - (-1)) = 2.$$

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13. Evaluate $\int_0^\pi x \sin x \, dx$.

Solution:
Using integration by parts:

$$\int x \sin x \, dx = -x \cos x + \int \cos x \, dx = -x \cos x + \sin x.$$

Evaluating from $0$ to $\pi$:

$$[-\pi \cos \pi + \sin \pi] - [0] = [\pi + 0] - 0 = \pi.$$

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14. Evaluate $\int_{-a}^{a} f(x) dx$ for $f(x) = \frac{x}{x^2+1}$.

Solution:
Since $f(x)$ is an odd function,

$$\int_{-a}^{a} f(x) dx = 0.$$

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15. Evaluate $\int_0^\pi \frac{dx}{1 + \cos x}$.

Solution:
Using $1 + \cos x = 2 \cos^2 (x/2)$,

$$\int_0^\pi \frac{dx}{2 \cos^2(x/2)} = \int_0^\pi \frac{\sec^2(x/2)}{2} dx.$$

Solving gives $\pi$.

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PART 4: Challenging Problems

16. Evaluate $\int e^x \cos x \, dx$.

Solution (By Integration by Parts Twice):
Let $I = \int e^x \cos x \, dx$.
Using integration by parts twice,

$$I = e^x (\cos x + \sin x)/2 + C.$$

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17. Evaluate $\int \frac{x}{(x+1)(x+2)} \, dx$.

Solution (By Partial Fractions):

$$\frac{x}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}.$$

Solving for $A$ and $B$,

$$\int \left( \frac{1}{x+1} - \frac{1}{x+2} \right) dx = \ln |x+1| - \ln |x+2| + C.$$

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18. Prove that $\int_0^1 \frac{\ln(1+x)}{x} dx = \frac{\pi^2}{12}$.

Solution:
Expanding $\ln(1+x)$ into a series and integrating term by term, we obtain $\frac{\pi^2}{12}$.

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19. Evaluate $\int_0^\pi x^2 \sin x \, dx$.

Solution:
Using integration by parts twice,

$$\int x^2 \sin x \, dx = -x^2 \cos x + 2x \sin x + 2 \cos x.$$

Evaluating from $0$ to $\pi$,

$$\pi^2 - 2.$$

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20-25: Advanced problems on Gamma Function, Beta Function, and Improper Integrals.

These will involve tricky definite integrals, convergence tests, and improper integration techniques.