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Integrals

Long Answer Type

261.

Evaluate: $\int_{0}^{π} \frac{xsinx}{1 + \cos^{2} x} dx$

262.

Evaluate: $\int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx$

Short Answer Type

263.

Evaluate: $\int \sec^{2} (7 - x) dx$

264.

If $\int_{0}^{1} (3 x^{2} + 2 x + k) dx = 0$ , find the value of k.

Long Answer Type

265.

Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$

266.
Evaluate: $\int \frac{(x - 4) e^{x}}{{(x - 2)}^{3}} dx .$

$I = \int \frac{(x - 4) e^{x}}{{(x - 2)}^{3}} dx I = \int e^{x} (\frac{x - 2}{{(x - 2)}^{3}} - \frac{2}{{(x - 2)}^{3}}) dx I = \int e^{x} (\frac{1}{{(x - 2)}^{2}} - \frac{2}{{(x - 2)}^{3}}) dx$

Thus the given integral is of the form,

$I = \int e^{x} |f (x) + f' (x)| dx where, f (x) = \frac{1}{{(x - 2)}^{2}}; f' (x) = \frac{- 2}{{(x - 2)}^{3}} I = \int \frac{e^{x}}{{(x - 2)}^{2}} dx - \int \frac{2 e^{x}}{{(x - 2)}^{3}} dx = \frac{e^{x}}{{(x - 2)}^{2}} - \int \frac{e^{x} (- 2)}{{(x - 2)}^{3}} dx - \int \frac{2 e^{x}}{{(x - 2)}^{3}} dx + C So, I = \frac{e^{x}}{{(x - 2)}^{2}} + C$