Question

1. (5 marks) Find the indefinite integral using the given substitution $\int x^3 (2 + x^4)^5 dx$, $u = 2 + x^4$, 2. (5 marks) Using the given substitution find the exact value of $\int_{3\pi}^{4\pi} \frac{sec^2(1/x)}{x^2} dx$, $u = 1/x$. 3. (5 marks) Use integration by parts to find $\int_0^1 (x^2 + 1)e^x dx$. 4. (5 marks) Use integration by parts to show that $\int x^n e^x dx = x^n e^x - n \int x^{n-1} e^x dx$, where n is a positive integer. 5. (5 marks) Using partial fractions calculate the indefinite integral $\int \frac{x^2 + 2x - 1}{x^3 - x} dx$. 6. (5 marks) Using partial fractions calculate $\int_0^1 \frac{x - 1}{x^2 + 3x + 2} dx$.

          1. (5 marks) Find the indefinite integral using the given substitution
$\int x^3 (2 + x^4)^5 dx$, $u = 2 + x^4$,
2. (5 marks) Using the given substitution find the exact value of
$\int_{3\pi}^{4\pi} \frac{sec^2(1/x)}{x^2} dx$, $u = 1/x$.
3. (5 marks) Use integration by parts to find
$\int_0^1 (x^2 + 1)e^x dx$.
4. (5 marks) Use integration by parts to show that
$\int x^n e^x dx = x^n e^x - n \int x^{n-1} e^x dx$,
where n is a positive integer.
5. (5 marks) Using partial fractions calculate the indefinite integral
$\int \frac{x^2 + 2x - 1}{x^3 - x} dx$.
6. (5 marks) Using partial fractions calculate
$\int_0^1 \frac{x - 1}{x^2 + 3x + 2} dx$.

$1. (5 marks) Find the indefinite integral using the given substitution ∫ x^3 (2 + x^4)^5 dx, u = 2 + x^4, 2. (5 marks) Using the given substitution find the exact value of ∫3π^4π(sec^2(1/x))/(x^2) dx, u = 1/x. 3. (5 marks) Use integration by parts to find ∫0^1 (x^2 + 1)e^x dx. 4. (5 marks) Use integration by parts to show that ∫ x^n e^x dx = x^n e^x - n ∫ x^n-1 e^x dx, where n is a positive integer. 5. (5 marks) Using partial fractions calculate the indefinite integral ∫(x^2 + 2x - 1)/(x^3 - x) dx. 6. (5 marks) Using partial fractions calculate ∫0^1 (x - 1)/(x^2 + 3x + 2) dx.$

Added by Evelyn E.

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